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COMPUTATION RULES AND LOGARITHMS 



•» 



Jl^^m 



COMPUTATION RULES 



AND 



LOGARITHMS 



WITH TABLES OF OTHER USEFUL FUNCTIONS 



BY 

SILAS W. HOLMAN 

Peofessob op Physics at the 

MASSAOHnSEITS iHSTITUTB OF TEOmfOLOST 



MACMILLAN AND CO. 

AND LONDON 

1896 

All rights reaevoed 



7 i 



/\-%%Q>35' 



Copyright, 1895, 
By S. W. HOLMAN. 



NottoDoB 33rtZ3 

J. S. Gushing & Co. - Berwick & Smith 
Norwood Maes. U.S.A. 



CONTENTS. 



PAGE 

Peefacb vii 

Computation Kules xi 

Proper Number of Places of Significant Figures xi 

Fundamental Principles xii 

Kules in Detail xiii 

Rejection . . xiii 

Multiplication and Division xiii 

Logarithms xiii 

Addition or Subtraction xiv 

Nujnerical Substitution in FormulaB xiv 

Notation by Powers of Ten xv 

Statement of the Method xv 

Symmetrical Grouping of Figures xvi 

Examples i to ii xvii 

Logarithms. Nature of xxiv 

Tables, Description of xxv 

To Find the Logarithm of a Number xxvii 

Decimal Point in Logarithmic Tables xxix 

Antilogarithms : Number Cokresponding xxx 

Cologarithms .... xxxi 

Habit in Reading off Numbers and Logarithms xxxii 

PovsTERs and Roots by Logarithms xxxiv 

Powers and Roots of Numbers greater than Unity xxxiv 

Powers of Decimal Fractions xxxiv 

Roots of Decimal Fractions xxxv 

Squares and Square Roots xxxvi 

Reciprocals xxxviii 

Natural Sines, Cosines, Tangents, and Cotangents . . xxxviii 

V 



VI CONTENTS. 

PAGE 

Logarithms of Sines, Cosines, Tangents, and Cotangents . xxxix 

Slide Wire Ratios xl 

Definitions and Explanations underlying the Computation 

Rules xli 

Significant Figures xli 

Places of I'iguies xli 

Places of Decimals xlii 

Accuracy; Reliability xlii 

Mean ; Average xlii 

Deviation Measure xlii 

Rules for Significant Figures xliii 

Rejection Error xliii 

Law and Amount of Accumulated Rejection Error xliv 

TABLES. 

Logarithms, Four-Place ' 2 

Antilogarithms, Four-Place 4 

COLOGARITHMS, FoUR-PlACE 6 

Logarithms, Five-Place 9 

Square Roots and Squares, Four-Place 30 

Reciprocals, Four-Place 34 

Slide Wire Ratios, Four-Place .36 

Natural Sines and Cosines, Four-Place 38 

Natural Tangents and Cotangents, Four-Place 40 

Logarithms of Sines and Cosines, Four-Place 42 

Logarithms of Tangents and Cotangents, Four-Place ... 44 

Logarithms of Sines, Cosines, Tangents, and Cotangents, 

Five-Place 47 

Constants 70 



PREFACE. 



It -would probably be within safe limits to assert that one-half of 
the time expended in computations is wasted through the use of an 
excessive number of places of figures, and through failure to employ 
logarithms. . This waste might be almost wholly avoided by follow- 
ing a few simple computation rules and practising slightly with 
logarithm tables. 

The loss from the use of superfluous figures will be appreciated 
when it is considered that in direct or logarithmic multiplication and 
division with four, five, and six places of figures the work is respec- 
tively in the ratio of i : 2 : 3, or perhaps more nearly 2:3:4. Thus 
contrary to the fallacious excuse so commonly given that it is just 
about as easy to use six- or seven-place tables as smaller ones,' the 
work is doubled or trebled by the use of six places instead of four. 
Even the employment of six- or seven-place tables, and dropping 
superfluous places when four or five are desired, causes much loss 
of time. 

The proper employment of logarithms for work of four or more 
places effects a saving of one-quarter and upward of the time 
required for direct multiplication or division, with a lessening of 
fatigue and a gain of accuracy. 

The following pages contain simple rules to enable one to answer 
for himself the question, how many places of figures ought I to use 
in this computation ? — also, an explanatioii of the use of the nota- 
tion by powers of ten; certain instructions, more or less novel in 
form, as to the use of the logarithm and other tables ; and a collection 
of useful tables. This collection is designed to contain all the mathe- 
matical tables ordinarily required, and nothing more, in practical 
work in all branches of the engineering professions, and by students 
of physics, chemistry, and engineering, for work of any grade not 
exceeding about one-twentieth of one per cent in accuracy. For 



Vin PKEPACE. 

many persons the present volume should, therefore, provide all the 
logarithmic and trigonometric tables needed for the entire range of 
their practice. For work of greater precision than the above limit, 
the more bulky Vega, or some similar reliable seven-place table 
would be reqmred. It is exceedingly rare that more than six or 
seven places are necessary, while for most work five are sufficient, 
although a striking chapter of absurd illustrations might be gleaned 
from various text-books and tables where ten- and even twenty-place 
logarithms are given, often for quantities uncertain in their fourth 
or fifth place. Person^ doing much work with squares, cubes, square 
roots, cube roots, or reciprocals of more than four places would natu- 
rally make use of the Barlow Tables. 

The rules for significant figures (pages xi to xv) are intended to 
be terse, direct, and simple, so that they may be easily acquired and 
retained. The strong type emphasizes the leading portions. The 
ordinary and finer types give details and explanations. Tor the sake 
of affording still greater prominence to the main working portions, 
some explanatory matter which will be unnecessary to many per- 
sons has been transferred from its more logical position of precedence 
to the latter part of the text. These rules in various forms have 
been in successful use by large classes of students, in connection 
with the author's " Physical Laboratory Notes " (printed, but not 
published, by the Massachusetts Institute of Technology), and his 
" Precision of Measurements." The recognition of the need of such 
rules amongst engineers and others whose practical work demands 
rapid and reliable computations was the cause of their general intro- 
duction into this laboratory instruction. It is therefore hoped that 
they may render effective service to others besides the students for 
whom they have been more directly designed. 

In the arrangement of the tables, the effort has been exerted to 
make them correct, legible, systematic, and convenient in use. A 
new set of tables is, of course, liable to contain mistakes ; notices of 
errata will therefore be thankfully received. 

The special indexing of the corners of pages, the use of heavy 
type at points to be made conspicuous, the employment of spaces 
rather than rules for the partition of lines and columns, and the 
style of type, and kind of paper used, are believed to conduce to 
legibility. As to system of arrangement, there are few novelties 
other than the insertion of the logs, cologs, and reciprocals of i .000 
to 1. 100 at the top of the respective four-place tables, and the division 



PEBFACB. IX 

of most of the four-place tables so that the second page begins with 
5.0 instead of the customary 4.5. The frequent occurrence of cor 
rection and reduction factors, ranging from i.o to i.i, renders this 
by far the most frequently used part of the table ; while at the same 
time, on account of the large tabular differences, interpolation is 
here the most laborious. The insertion of logs, cologs, and recipro- 
cals from 1.0 to I.I with increments of o.ooi and o.oooi, respec- 
tively, in the four- and five-place tables, obviates this interpolation. 
In tables of antilogs and square roots the addition would be of little 
service. In the tables of logarithms and of square roots, heavier 
type has been used at apparently scattered points throughout the 
body of the tables. These points, in -the five-place logarithm tables, 
for instance, are where the first two figures in the mantissa change 
by one unit in the second place, e.g. 00, 01, 02, etc. The obvious 
service of this is to aid the eye in finding any desired mantissa in 
working the table backward to obtain the antilog or number corre- 
sponding. The object is, of course, the same in the other tables. 

As to the wisdom of departing from the usual custom of omitting 
decimal points entirely from logarithm tables, the author believes 
that the retention of the point promotes clearness of comprehension 
of the tables by beginners, and lessens mental effort in more experi- 
enced computers, especially when associated with the notation by 
powers of 10, as in the explanations here given. It seems unfortu- 
nate that this simple notation, so useful in computation and so great 
an aid in the explanation of numerical relations, is not universally 
incorporated into arithmetical instruction. 

The rules for the employment of logarithms and of the tables 
have not been prepared especially to meet the need of those entirely 
unfamiliar with the principles of logarithms, although they would 
probably be intelligible to any mature beginner. It is thought, how- 
ever, that the explanations and instructions given may prove an aid 
even to those who are already somewhat familiar with the subject. 

RoGBBS Laboratory of Physics, 

Massachusetts Institute of Technology. 

Boston, August, 1895. 



COMPUTATIOIN^ EULES. 



o»{o 



PROPER NUMBER OF PLACES OF SIGNIFICANT FIGURES. 

The following three pages contain the rules and their underlying 
principles in a condensed form for ready reference. For readers to 
whom some of the terms employed are unfamiliar, or who desire 
fuller proofs and explanations, some additional pages of " Definitions 
and Explanations " have been appended. 

These rules should enable a computer to decide at the outset of 
his work, or at the successive stages of it, what number of places of 
significant figures he should retain in order to avoid waste of labor 
on the one hand or sacrifice of accuracy on the other. They provide 
for a sufficient number of places to assure that (barring mistakes) 
the accumulated error arising from the rejection of further places 
shall be always smaller, usually much smaller, than the supposed 
uncertainty of the data or result, in computations involving not 
more than about 20 rejections. The retention of more places is 
worse than useless. It adds nothing to the accuracy of the result, 
although increasing materially the labor of computing, and the lia- 
bility to mistake. The aggregate value of the time thus wasted, 
— obvious enough to any one who has had occasion to perform 
extended computations, — may be appreciated from the fact that the 
use of five places where four would suffice, nearly doubles the labor ; 
using six places instead of four, nearly trebles it ; thus wasting 100 
and 200 per cent respectively of the necessary amount of work, and 
probably a greater proportion of time. Moreover, incongruities in 
the use of places of figures arouse skepticism as to the competence 
of the worker in other directions. 



Xil COMPUTATION EULBS. 

FUNDAMENTAL PRINCIPLES. 

Retain everywhere enough places to correspond to two unreliable 
places in the final result ; the direct object of this is to keep the first 
place of unreliable figures in the final result substantially free from the 
accumulated rejection errors. 

Exceptions. — A final result is seldom stated to more than one 
uncertain place unless the uncertainty of that place is small (say 
plus or minus four or less). 

Example : i, page xvi. 

Single direct measurements generally yield numbers extending to 
only one uncertain place. This should not, however, be taken as a 
reason for relaxing the application of the above rule to subsequent 
steps of the computation, especially in deducing the mean or average 
of several single observations. 

Final zeros occurring in decimal fractions should be retained 
when any other digit in the same place would be retained. This is 
of course essential to show that this place is known. 

The foregoing principles consistently carried out constitute en- 
tirely sufficient rules. But more detailed instructions are usually 
required at the outset. These are readily understood in view of the 
two following propositions, which one can easily verify by algebra 
or by numerical examples. 

Proposition I. — In multiplication or division, the percentage 
accuracy of the product or quotient cannot exceed that of the factor 
whose percentage accuracy is least. 

Proposition II. — In addition or subtraction, the result cannot 
be accurate beyond the first decimal place which is inaccurate in any 
component. 

A more general form of statement from which these follow is : 
If several numbers are multiplied or divided, a given percentage 
error in any one of them will produce the same percentage error 
in the result. If several numbers are added or subtracted, a given 
error or change in the digit in any decimal or other place will 
produce an. equal error or change in the digit in the same decimal 
place in the result. 



COMPUTATION RULES. XUl 



RULES IN DETAIL. 

Eejectioi^. — In casting off places of figures, increase by i the last 
figure retained when the first left-hand rejected figure is 5 or greater ; 
otherwise leave it unchanged. 

Example. — If the last two figures are rejected 

756827.9 becomes 756830. 

and 0.00 263 439 becomes 0.00 263 4. 

A MEAN OK AVERAGE should always be carried to two unreliable 
figures. 

A mean is more reliable than the single observation from which 
it is computed (in proportion to ^^/n, the square root of the number 
of observations). Thus, as the data frequently extend to only one 
unreliable figure, the mean will often have to be carried two places 
further than the single observation. 

Multiplication or Division. — Ascertain from the object of the 
work the percentage accuracy desired in the final result ; or, by inspec- 
tion of the data, find the percentage accuracy of that factor for which 
this is least, i.e. for which the deviation-measure or the estimated 
error, expressed as a per cent, is largest. See Example i, page xvii. 

In direct multiplication or division retain in every factor, product, 
and quotient throughout the entire process, and in final results, for an 
accuracy of about 

One per cent, or worse, four (4) places of significant figures; 

One-tenth per cent, or worse, five (5) places of significant figures; 
and so on. 

In the ordinary and the shortened processes of "long multi- 
plication," it is best to carry out the partial products one place 
beyond that yielding the last place required in the result under the 
above rule. 

Examples: 2-5, page xvii. 

Logarithms. — If the multiplication or division is performed by 
means of logarithms, the mantissa should contain as many places as 
are required by the foregoing rules for the direct process; i.e. for about 

One per cent, or worse, use four (4) place tables; 

One-tenth per cent, or worse, use five (5) place tables. 

Examples : 2-5, page xvii. 



xiv COMPUTATION RULES. 

Addition ob Subtkaction. — Ascertain from the stated object of 
the work the percentage accuracy desired. If this is about 

One per cent, or worse, carry the result to four (4) places of signifi- 
cant figures; 

One-tenth per cent, or worse, carry the result to five (5) places of 
significant figures; and so on, and carry each component quantity to 
that place of decimals which would correspond to this required place in 
the result, that is, stand in the same column with that place. 

Examples: 6-8, page xix. 

When the desired acoukacy is not stated, inspect the data 
to find the component whose first uncertain place is furthest to the left, 
i.e. whose deviation measure (page xlii), in units, not percentage or 
fractional, is greatest. Retain this component to two uncertain 
places, and all other components to the place which would stand in the 
same column with this second place. 

Examples: 6-8, page xix. 

N. B. — If the number of components approaches 20, care may 
well be taken in refined work that an unusually large rejection error 
does not enter through a special combination of rejected figures. 
The rules are, however, sufficient for the worst possible case. 

The computer should notice that the percentage precision of a 
result which is the difference of two or more quantities will usually 
be smaller, and may be much smaller, than that of any of the 
component quantities. 

Numerical Substitution in Formulae. — A large number of formulae 
may be represented by the type 

a-b ±c-d± ••• 
x = - 



p-q ±r-s ± 



where a, b, c, d, etc., represent numbers to be multiplied, divided, 
added, or subtracted, etc., as indicated. Any one or more of the 
factors and terms may be wanting ; or, there may be several in place 
of two ; and so on. 

Obviously, in order that the result x shall be accurate to a speci- 
fied per cent, both numerator and denominator must be at least of 
that accuracy, and each should therefore be carried out to the num- 
ber of places of significant figures needed in x. Then as the numer- 
ator consists of two or more terms ab and be added or subtracted, 



COMPUTATION KULES. XV 

inspection under the foregoing rules for addition or subtraction will 
show to what decimal place each of these terms must be carried. 
Further, a and 6 must each be carried to the number of significant 
figures thus required in the product ab, and so on. 

In complicated formulae this process of inspection is sometimes 
slightly troublesome, but is essential unless the necessary precision 
of the components has been otherwise studied ; as, for example, by 
the simple applications of the differential calculus as in the author's 
" Precision of Measurements." 

Examples: 9-1 1, page xx. Practical examples of substitution in 
moderately simple formulae. 

NOTATION BY POWERS OF TEN. 

Statement of the Method. — Eegard the decimal point as merely 
an affix whose sole purpose is to indicate which is the units' place 
of figures. Fix the attention firmly upon the units' place as the centre 
of symmetry of our customary system of notation. The too universal 
reference to the decimal point, rather than to the units' place, in 
arithmetical rules and explanations, has resulted in masking this 
symmetry and in thus depriving the student of its important aid. 
In oiu: common decimal system of notation, a digit in the units' 
place represents so many times unity, i.e. so many times 10° (=1), 
or so many units. In the first place to the left of the units' place 
the digit represents so many times 10+^, i.e. so many tens, and in the 
first place to the right, so many times io~', i.e. so many tenths; in 
the second place to the left so many times 10+^, i.e. hundreds, and to 
the right so many times io~^, i.e. hundredths; in the sixth places, so 
many times 10"'"° and io~^, i.e. millions and millionths, respectively; 
and so on. The fundamental symmetry of the whole system about 
the units' place is thus obvious, and should not be lost sight of. 

In counting up places, whether to right or left, always begin with 
the units' place as zero. 

It is clear, then, that we may write numbers in this way : 



for 90 


write 


9-10^; 


for 60G0 


write 


6.ooo'io' or 6" 10^ as the case may require ; 


for 345 


write 


S^S'io''; 


for 0.00 005 


write 


5-10-'; 


for 0.00 468 9 


write 


4.689-10-'; 


for 850.72 


write 


8.507210^ ; and so on. That is, 



Xvi COMPUTATION EULES. 

Separate the number into two factors, the first being the original 
number with the decimal point changed in position so as to follow the 
first figure; the other being lo-", where the sign is plus for a whole 
number and minus for a fraction, and where n is the number of places 
the decimal point has been moved. 

To transform a number expressed in tMs way back into the ordi- 
nary form, move the decimal point n places, making the number a 
whole (or a larger) number if n is plus, and a fraction if n is minus. 

Associate firmly in the mind the plus sign with whole numbers, 
the minus sign with fractions ; thus avoiding confusion as to the 
sign of n. 

In much work, the factoring need not be written out, but may 
merely be mental. 

This notation reduces the error and work of locating the decimal point in 
multiplication or division, especially in expressions containing several terms 
in the numerator and denominator. It is very helpful in connection with the 
characteristic of logarithms, and the location of the decimal point in evolution, 
involution, and finding reciprocals. It saves space and promotes clearness in 
expressing large numbers or small fractions, and it is the best aid in following 
the decimal point while using the slide rule. It also enables one to dispense 
with characteristics in certain parts of computations (see Examples, page xxi). 

An abbreviated notation helpful in one's own work, but perhaps not to be 
urged for general adoption, consists in dropping the -lo, thus, 

instead of 4.507-102 write merely 4.507^ 
instead of 5.3704-10-3 write merely 5.3704-3 

The adoption of the bracket or parenthesis, e.g. (4.507)2, for either notation 
in cases of possible doubt removes all risk of mistaking these indices for ordinary 
exponents of powers. 

Examples 9, 10, 1 1 give incidentally illustrations of the use of the notation 
by powers of 10. 

Symmetrical Grouping of Figures. — For writing numbers, adopt 
the following system of groups and spaces : — 

Write 143 258.64 796 

instead of 143,258.647,96, the usual method. 

A still clearer method would be to write 

143 25864 796 

denoting the units' place by the heavy figure, but this is impracti- 
cable. The proposed system is symmetrical about the units' place, 
the customary system is not. It groups together the units, tens, 
and hundreds of thousandths, of millionths, etc., as well as of thou- 



COMPUTATION EULES. XVU 

sands, millions, etc. It is clearer and less liable to error by the 
substitution of spaces for the commas to mark off the groups 
Exception is usually to be made in the case of a decimal fraction 
containing only three or four places. Thus write 0.4612 rather 
than 0.46 12, and 6.382 rather than 6.38 2. 

EXAMPLES. 

Example 1. — Suppose that a final result was stated as 298549. ± o.io per 
cent; this would mean that its uncertainty or deviation-measure or estimated 
measure of accuracy (see page xlii) was ± o.io per cent. To how many places 
should it be retained? o-i per cent of the number is 0.00 1 x 300000= 300. 
Therefore the last three places are uncertain, but as the uncertainty in the 
first left-hand one is small (3) , two uncertain places should be retained. The 
result, therefore, should be written 298 550. -± o.io per cent. 

Suppose a result given as 47.58 243 5 ± 0-0062- This would be an incorrect 

use of figures. The ± 0.0062 shows that the result is uncertain in the third and 

fourth, and therefore in all subsequent decimal places.* The fifth and sixth 

places of significant figures are thus unreliable, so that the seventh and eighth 

places are entirely valueless, and should, therefore, be rejected. We should 

be at liberty to -use our judgment as to whether the result should then be 

written 

47.58 24 ±0.00 62 or 47.582 ± 0.006, 

since the uncertainty in the fifth place is large. The second is more common 
practice. In this example the uncertainty is ± 0.0062/47. =± 0.00013, °^ 
±0.013 Psr cent. It might, therefore, have been expressed as ± 13 parts in 
100,000, or ± 0.013 per cent instead of as ± 0.0062 units. It is always expressed 
in the same unit as the quantity itself, e.g. ft., lbs., etc., except when directly 
stated to be a percentage. 

Example 2. — Desired with an accuracy of 2 per cent, the volume of a right 
circular cylinder whose radius r is 6-0428 inches, and length I 12.653 inches- 
Volume V= irrH. By the rule for multiplication, since the result is desired to 
worse than one per cent, the data and all steps should be carried to four places 
of significant figures- Hence, we should have 

V= 3-142 X 6.0432 X 12.65 = I4SI' 



By ordinary multiplication : 



By shortened multiplication : 



6.043 
6.043 
I8I29 
24162 

36258 


3.142 
36.52 
6284 
1 5710 

18852 
9426 
1 14.7 


"4-7 
12.65 

S73S 
6882 

2294 
1147 

145 1. 


6.043 

6.043 

36258 

240 

18 


3.142 
36.52 
9426 
18 84 

I SS 
6 


114.7 
1265 

"47 
2294 
690 


36.52 


36.52 


55 




"4-7 


1451- 



* This quantity ± 0.0062, or whatever may be its value, is the " average devia- 
tion" or the "deviation-measure" of the result; that is, the average amount by 
which several results similarly obtained would differ from their mean. A fuller 
explanation is given at page xlii. The " probable error," which is nearly identical 
with the average deviation, is commonly used in its stead. Either suffices. 



XVIU 



COMPUTATION EXILES. 



Observe that the partial products beyond the place standing over the fifth 
place of the result In each multiplication are useless. Hence the obvious saving 
of labor in the shortened process, which is also more compact. The process is 
easily understood by inspection of the example. Multiply first by the first left- 
hand figure of the multiplier. If the resulting partial product has one more 
place than is desired in the result, then drop the last figure of the multiplicand 
when multiplying by the second figure of the multiplier ; drop the last two, when 
multiplying by the third figure ; and so on. If, however, the first partial product 
has not the desired number, the dropping of figures must be deferred till the 
third figure of the multiplier is used. 

Xizample 3. — Desired the volume V=TrH of a right circular cylinder 
whose dimensions are 

in. in. 

r = 6.0428 ± J per cent, I = 12.653 ± -f^ per cent. 

The result cannot be more accurate than the least precise factor, which is 
obviously r. Under the rule, I per cent computations call for five places of 
significant figures. Hence we should have to find by multiplication or by five- 
place logarithms, 

V= 3.1416 X (6.0428)2 X 12.653. 

Note in this connection that the error in V is proportional to twice the 
percentage error in r, since r enters in the second power, and that in general 
where a numberis raised to any power n the percentage error in the result is 
increased to n times its value in the data. These rules, however, provide suflS- 
ciently for such cases. See " Definitions and Explanations." 

Example 4. — Desired the volume V = irrH of a right circular cylinder 
whose dimensions are given as 

r = 6.043 ± 0.017 inches, I = 12.653 ± 0.038 inches. 

Under the general principle of retaining places to correspond to two uncer- 
tain figures in the result in the data, r should have four places and I five places, 
judging from their stated precision. But the weaker quantity fixes the number 
of places in the result, so that we should use but four places : 

F= 3.142 X (6.043)2 X 12.65. 

Ezample 5. — Desired the ratio of the diameter to the length in each of the 
Examples 2, 3, and 4. 

The number of places of figures to be used would be respectively four, 
five, and four, just as in the above solutions. A factor in the denominator fol- 
lows precisely the same rule as to places of significant figures as the one in the 
numerator. To contrast the ordinary and shortened solutions, the following are 
given : 



12.65)6.043(0.4777 
5060 
9830 
8855 
9750 
8855 
8950 
8855 



6.043 
5060 
9830 

8855 

975 
889 

"86 

91 



12.65 



0.4777 



COMPUTATION RULES. XIX 

Iizample 6. — Desired the value to 0.6 per cent of 

47-34 89 + 174.32 825 - 5.62 147. 

For o. 1 per cent or worse (page xiv), we must retain places in tlie compo- 
nents to correspond to five places in the result. By inspection we see that the 
result will be slightly more than 200. Hence its fifth place will be the second dec- 
imal place, and we need retain no place beyond that in the components. Thus, 

47-35 
174-33 

221.68 

-s-62 

216.06 l± an unknown amount as precision-measure]. 

Example 7. — Desired the algebraic sum of 

47'- 34 89 ± 0.0042, i74'.32 825 ± 0.00 089, and — 5'.62 147 ± 0.00 008. 

By inspection the weaker component, that is, the one whose first uncertain 
place is furthest to the left, is the first number. Retaining this to two uncertain 
figures would carry it to the fourth decimal place. It will then be useless to 
retain the other components beyond that place, and we shall have 

47-3489 
174-3283 



221.67 72 
-S-621S 



216.0557 [± more than o'.oo 42]. 

Example 8. — Desired the algebraic sum of 

47'. 34 89 ± 0.05 per cent, 174'. 32 825 ± 0.02 per cent, — 5'.62 147 ±0.1 per cent. 

0.05 per cent of 47. is 0.024 i 0-02 per cent of 174. is 0.035 > °- ' P^'^ '^^^^ °^ 
5.6 is 0.00 56. Hence the weakest component is now the second, and this, and 
consequently the others, should be retained to three decimal places. Thus, 
we have 

47-349 

174-328 

221.677 

. —5.621 

216.056 [± more than o'.035]. 

Example 9. — The horse-power, HP, which could safely be transmitted by 
a wrought-iron shaft of diameter d inches, running at a speed N rotations per 
minute, the safe shearing load of wrought iron being represented by/, is given 
by the expression 

^j,_ 2,r^yjy 

16-I2-3300O 



XX COMPUTATION KTJLES. 

[Deduced from Lanza's "Applied Mechanics," page 336. The several con- 
stants 2, 16, 12, and 33000 would of course be combined into a single constant 
in a working formula, but they are here left separate for purposes of better 
illustration.] 

To how many places of significant figures should the quantities, result, and 
various steps of the computation be carried out to assure against a computation 
error in the result, sensible as compared to one per cent ? 

Solution. — In this and all similar problems, where the expression consists 
merely of a number of factors in the numerator and denominator (either or 
both), without additions or subtractions, the solution of the significant figure 
problem can be made without any knowledge of the magnitude of the component 
quantities, such as d, f, y, etc. In this example, as the result is desired to one 
per cent, according to the rules it should be carried to four places of significant 
figures. Hence, according to the rule, page xiii, or to Proposition II, page xii, 
each factor of the whole expression should be carried to four places. In this 
expression every quantity is a factor, either in the first or a higher power, viz. 2, 
ir2, d^, /, JV, 16, 12, and 33000. Each, therefore, should he carried to four 
figures. Hence, also, if direct multiplication be employed in the solution, each 
product and quotient must be carried to four places. If logarithms are used 
(they should be) four-place tables should be chosen. When a quantity enters 
as a factor of the nth power this is equivalent to its entering n times as a simple 
factor or as n separate factors, all with the same percentage error of the same 
sign. See also note under example 3. 

The constants 2, 16, 12, and 33000 do not require to be carried to more 
places than they are here given because they are complete as they stand, that is, 
all further figures are known to be zero as a matter of definition or mathematical 
fact. If either of them had been an experimental constant , that is, determined 
by measurement, it should have been carried out to four places even if the last 
figure or two were zero. For instance, if experimental, the 16 should have been 
written 16.0, 16.00, 16.000, and so on according to the number of places to which 
it was known (see rule, page xii). Failure on the part of those who write such 
formulse to adhere to this convention, or to indicate in some clear way the 
degree of accuracy possessed by such constants, is a serious source of annoyance 
and trouble to those who use them. 

As elsewhere it must not be inferred if certain of the quantities, e.g. d or/, 
in this expression cannot be carried out to this desired number of figures, that 
consequently the result will not have the accuracy desired in the given case. 
The outcome of such a condition would merely be that these factors would be 
liable to introduce more than a safe computation error. For instance, if /were 
given as loioo lbs. per square inch, we should have no certainty that it was 
carried far enough. The presumption would be that it was cori-ect to but three 
places, and therefore not exact enough. If, however, from a knowledge of tlie 
subject we were aware that the best known value was loi 10, we should know 
that the error from using loioo was only i in 1000 or o. i per cent, and hence 
admissible. On the other hand, if we know that the best value was 10050, we 
should know that the computation error in tlie result from using loioo was 0.5 
per cent, and hence by no means safe in the above problem. 

More complete methods for ascertaining the exact accuracy needed in each 
component measured quantity in such formulae, are given in the author's 
" Precision of Measurements." It is to be remembered that we are now dealing 



COMPUTATION KULES.- 



XXI 



merely with rules for computation errors, and these are not suited to the other 
problem. They are intended to secure a safe number of places for the worst 
case, and would, therefore, impose unduly severe requirements as to the accu- 
racy necessary in the measurement of the components in most cases. 

Numerical Substitution. — The numerical expression to he solved if written 
out would he 



2-3. 142^- 1. 3642- 10 000-300 

1 6- 12- 33 000 
2-3.1422-1. 3645- 10^-3-102 . 



in the ordinary notation ; 



i.6-ioi'i.2-ioi-3.3-io* 
2(3.142)^(1.364)810^-32 



in the notation by powers of 10 (page xv); 



1.61-1.21-3.3* 



in the abbreviated notation by powers of 10 (page xv). 



The first would be worked out in the usual manner by direct multiplica- 
tion or by logarithms, as shown below. 

The second would be worked as follows : 
Multiply together the terms other than 10" of the 

numerator, i.e. 2 x 3.1422 x 1.364'^ X 3 = 150.3 

Multiply together the terms other than 10" of the 

denominator, i.e. 1.6 x 1.2 x 3.3 = 6.336 

Divide numerator by denominator = 23.72 

Add together all indices of powers of 10 in numerator, also in denom- 
inator, and subtract the latter from the former. Or, better, add 
(algebraically) all the indices, reversing the sign of those in the 
denominator, thus : 4 + 2 — 1 — 1 — 4 = 0. 
The result is therefore 23.72'io'' = 23.72 

Note distinctly that all this writing out of the fraction and of the several 
steps is merely for the purpose of this explanation. In an actual solution such 
of these operations as are essential to the work would be conducted mentally, 
the actual multiplication and division alone being written out. 

If the solution were made by logarithms, it might assume either of the two 
following forms. The first is the usual one, the second shows how the use of 
the powers of 10 enables us, if we choose, to dispense with writing character- 
istics in very many places, —a saving of just so much labor. The factor 10" 
in the second result is, of course, obtained by summing mentally the indices of 
the factors 10, those in the denominator being taken with reversed sign, as in 
the preceding paragraph. These indices would not be written out, but taken 
by direct inspection of the data as originally written. 



Denorainator. 


Usual 
Method. 


Dropping 

Characteristics. 


Numerator. 


Usual 
Method. 


Dropping 

Dharacterist 


log 16. 


= 1. 2041 


.2041 


log 2. 


= 0.3010 


.3010 


log 12. 


= 1.0792 


.0792 


2 X log 3.142 


= 0-9944 


-9944 


log 33 000. 


= 4-5185 


•5185 


3 X log 1.364 


= 0.4044 


.4044 


log denom. 


= 6.8018 


.8018 


log 10 000 
log 300. 


= 4- 

= 2.4771 


■4771 








Result, 


8.1769 
6.8018 


2.1769 
.8018 




1-3751 
23.72 


I-3751 
23.72-1 



XXH COMPUTATION RULES. 

Example 10. — The crushing load of a hollow, cast-iron pillar of circular 
section, with concentric surfaces of diameters D and d as given by Hodgkiuson 
(Lanza, "Applied Mechanics," page 332) is 

„ ,r(D2 - (22) 

c = 100 801—^^ -■ 

4 

Desired to ten per cent the load which a pillar of external and internal diame- 
ters 4.032 inches and 2.16 inches, respectively, would carry. How many places 
of figures should he used in the computation ? 

Ten per cent results call for three figures in all factors (page xiii). The 
factors in this expression are 109 801, ir, {D^ — d^), and 4, each of which 
should therefore be carried to three figures. The first two should therefore be 
1 10 000 and 3. 14. The 4 is a complete number as it stands. D''—d^=4.d^—2.2^ 
roughly = 16.0 — 4.8 = 11.2. To have three figures, it should therefore be car- 
ried to the first decimal place. Then as it is made up of two quantities, one 
subtracted from the other, each of these should be carried (page xiv) to the 
decimal place desired in the result, i.e. to the tenths' place. This requires D^ 
to contain three figures, 16.0, and hence D should contain three figures (since 
D^ consists of the factors DxD), i.e. should be written 4.03 inches. The require- 
ment of one decimal place in d^ calls for but two figures, 4.8, and hence two 
figures, 2.2 inches, in d. The numerical expression to be solved would then be 

3.14(4.032 _ 2.2^) 

c= 1 10 000 -^ ^^^ ^ -, 

4 

which would be most easily worked by a simple slide rule, or by direct multipli- 
cation. 

Example 11. — Desired to o. i per cent the fraction of dry steam in a 
sample of steam, using the following observations made with the "Barrus 
Calorimeter" (Peabody's " Themodynamics of the Steam Engine," page 234), 
the formula being 

_ TF(g2 -qi)+e- w(q - qs) 
*- wr 

where 

X = fraction of a mixture which is dry steam . . ... 

JF = weight of cooling water 573 5 lbs. 

w = weight of condensed water 29.89^3. 

t = temperature of steam 

(1 = initial temperature of cooling water 37°-49 ^• 

f2 = final temperature of cooling water 83°.84F. 

<3 = temperature of condensed water 304'^. 88 F. 

qs = "heat of liquid " at t°s, from ig and tables . , . . . . 274.4 

q-i = "heat of liquid" at t°2, from t^ and tables 51.91 B. T. U. 

qi = " heat of liquid " at t°u from ti and tables ... . 5.53 B. T. U. 

5 =" heat of liquid " at «°, from f and tables 287.6 B. T. U. 

e = radiation loss during test 120. B. T. U. 

r = latent heat of steam at 2°, from tables 891.2 B.T.U. 

What number of places of figures should be used throughout ? 

Solution. — For o. i per cent the result, and therefore all factors, should 
have five places of figures (page xiii). The only factors of the whole expression 



COMPUTATION RULES. xxiii 

are the whole numerator, w, and r. They should therefore be carried to five 
places. Note, however, that w and r are not so given in the data. Whether, 
however, they are given closely enough must he determined by other means (see 
remark at foot of page xx). Their product must, however, be carried to five 
places, and five-place log tables should be employed. The numerator consists of 
three terms, whose values, roughly, are 

570(52. — 6.)= 2600., 120., and 30(288. — 274.)= 420. 
. •. numerator = 2600. + 120. — 420. = 2300., roughly. 

To have five places the numerator must be carried to one place of decimals, 
as also must each of its terms. The first term is composed of two factors, W 
and (32 — Si), each of which must therefore be carried to five places. Then 
36 (Z2 — <?i — 52. — 6 = 46. roughly, it must be carried to the third decimal place. 
Hence q-^ and 51 must each be carried to the third decimal place. The third 
term, in order to extend to the first decimal place, must contain four figures. 
It consists of two factors, w and {q — qa), each of which must thus contain four 
figures. To make the value oi q - qs contain four figures, its numerical value, 
14, must he carried to the second decimal place. This would require that both 
q and qs be carried to the second decimal place, or to five figures each. 

To summarize, then, putting in an interrogation point wherever a figure is 
wanting in the data, we have as the numerical expression to be solved 

573-S ? (51-91 ? - 5-53?)+ I20-? - 29.89( 287.6? - 274.4?) ^^ 

"" - 29.89 ?x 891.2? °-9^^- 

Obviously, then, on inspection, although we might cany out the products 
W(q2 — qi), io{q — qs), and lor, each to the necessary five figures, much doubt 
is cast upon the sufficiency of the data themselves to give the desired o.i 
per cent in the result. The problem would require a detailed study by other 
tnethods to decide that point, — the result of which, it may be incidentally 
asserted, would be adverse. 

Note. — It may not be amiss in connection with these problems to call 
attention to the very large amount of engineering computations in which four, 
often three (slide rule), places of figures are abundant. In the design of 
machines and structures, the strength and sizes of parts, such as struts and tie 
rods, beams, pillars, shafting, etc., and the strains or stresses in them, often can- 
not, or need not, be fixed upon within an accuracy of one or even many per 
cent. This limit is fixed by the unreliability of materials or workmanship, or 
by ignorance of the exact conditions to which the parts may be subjected. Like 
uncertainties affect many of the data upon which specifications, estimates, and 
contracts for engineering work are based, and the experimental constants in 
sundiy formulae. More than three or four places of figures can be indulged in 
for such work only at an extravagant waste of time. On the other hand it is 
necessary to discriminate sharply such operations as linear, angular, surface, 
and sometimes levelling, measurements in surveying and geodetic work where 
the accuracy may be very high. 



LOaARITHMS. 



oXKo 



Before reading the following pages become familiar with the 
"Notation by Powers of lo," page xv. 

The common or Briggs logarithm of a given number is the expo- 
nent of that power to which lo must be raised to produce the 
number. Thus, 3 is the common logarithm of 1000, since 10' = 1000. 

To multiply numbers together, add their logarithms. The sum is 
the logarithm of the desired product. 

Proof. — The product 10" x 10' X ••• X iC" is io''+'-+'", since 
powers of the same number may be multiplied together by adding 
their exponents. Therefore, if ^1=10°, iJ=io', •••, Jf = lo", 
^ X -B X ••• X M= io''+'+-+"'. That is, a + & H h m is the log- 
arithm of ^4 X -B X ••• y. M. But a, b, •■•, m are respectively log^, 
log B, •■■, log 31. Hence, log (A x B x ■■■ X M) = log J. -|- log B 
+ ■■■ +\o%M. 

To divide one number by another, subtract the logarithm of the 
latter from that of the former. The difference is the logarithm of 
the quotient. 

Proof. — Following the foregoing notation, A/B = lo'/io' = 
10° X 10' = lo""'. Hence log A/B = a—b = log A — log B. 

Logarithm of a number of several figures. — As 1 = 10" and 10 = lo'^, 
the logarithm of i is o and of 10 is i, and the logarithm of any 
number greater than o and less than 10, that is, of any number in 
the units' place (whether or not followed by a decimal fraction) is 
less than i, that is, it is a fraction. It is expressed as a decimal 
fraction. 

From the definition of a logarithm it is obvious that the logarithm 
of any stated power of 10 is the index of the power; i.e. log 10*" 
= ±n when ±n is any number, whole or fractional, positive or neg- 



LOGARITHMS. XXV 

ative. Hence, taking first a specific example, the logarithm of 306.2, 
being of course the same as the logarithm of its equal, is the same 
as the logarithm of 3.062-10^ (see "Notation by Powers of 10," page 
xv), -which is, log 3.062 + log 10^ = .4860 + 2., which is usually 
written 2.4860. The .4860 is found from tables, as shown later. 

Similarly, log 0.00 306 2 = log 3.062-10"^ = log 3.062 +log io~^ = 
.4860 — 3., which is usually written either 3.4860 or 7.4860 — 10, as 
will be further explained. 

The logarithm then consists of, or may be separated into, two 
parts, viz. first, the decimal part called the mantissa, which is the 
logarithm of the first factor in the above separation; second, the 
integral part, or whole number, preceding the decimal point, and 
called the characteristic or index, which is the logarithm of the 
second factor 10=*=", and which, therefore, is ±n. 

Tables of logarithms contain the logarithms of the numbers from 
I. to 10., by steps larger or smaller, and to as many decimal places as 
may be requisite for the accuracy sought in the work in which they 
are to be employed. But all numbers whatever, from o to 00, are one 
of these numbers i. to 10. multiplied by some power of ten, i.e. by 
^o-". Tor example. 



4 628 326 = 4.62 832 6-10^, and 0.03 986 = 3.986-10^ 



2 



Hence, the tables enable one, by merely prefixing to the tabular 
value the proper " characteristic " ± w, to obtain the logarithm of 
any number whatever, from zero to infinity. The quantity directly 
given in the table is obviously the mantissa of the desired logarithm, 
and is therefore always a decimal fraction. 

Since any table gives the mantissa to only a specified number of 
decimal places, it can represent only a correspondingly restricted 
number of places of significant figures in the original number. It is 
to be remembered that a change of one unit in the last decimal place 
of the mantissa corresponds at all points throughout a table to a 
constant percentage change (" Definitions and Explanations ") in the 
number corresponding. The amount of this change is such that it 
becomes the proper custom to use logarithm tables giving the mantissa 
to a number of places equal to the number of significant figures re- 
tained in the original quantities. Thus, if the numbers entering into 
the computation are properly retained to four places- of significant 
figures, a four-place logarithm table should be used in connection with 
them ; if to five significant figures, a five-place table ; and so on. 



XXVI LOGARITHMS. 

Four-place logarithm tables contain the logarithms to four decimal 
places of all numbers of three figures from i.oo to 9.99, and enable 
one by interpolation to obtain the four-place logarithm of any four- 
place number from i.ooo to 9.999. By merely prefixing the proper 
characteristic ± n, therefore, the four-place logarithm of any four- 
place number from o to 00 is obtained, or, in other words, the four- 
place logarithm of any number whatever from o to 00 in so far 'as 
this is governed by the first four significant figures of the number. 
Four-place tables should not be employed upon work of an accuracy 
exceeding one-half of one per cent. 

Five-place tables give directly the logarithms to one place further 
than four-place tables. I.e. to five decimal places, for numbers from 
1.000 to 9.999, and thence, by interpolation, from i.oooo to 9.9999. 
Thus, with the proper characteristic, these tables furnish the log- 
arithms of all five-figure numbers from o to 90, that is, of any num- 
ber whatever in so far as this is governed by the first five significant 
figures of the number. Five-place tables should not be employed 
in work of an accuracy exceeding one-twentieth of one per cent. 

Six-place tables are sometimes arranged with the same steps as 
five-place, i.e. giving directly the logarithm to six decimal places 
of numbers from i.ooo to 9.999 only. Such tables are of no prac- 
tical service ; for it is entirely useless to employ six-place logarithms 
in work on five-place numbers, and interpolation for six-place num- 
bers in tables of so large steps as this, besides being less reliable, 
is more laborious and annoying than is the use of the more bulky 
tables of smaller steps. 

If six-place tables are desired, it is usual to employ, dropping 
the last place, tables which give directly seven-place logarithms of 
numbers from i.oooo to 9.9999 with convenient interpolation tables 
for the next place. Of these, the Vega tables are among the most 
convenient, legible, and reliable, being also comparatively inexpen- 
sive. ■ The seventh place is very rarely demanded by physical, 
chemical, or engineering work. 

The relative labor in using four, five, and six place tables lies probably 
between the ratios 1:2:3 and 2:3:4. Assuming the first estimate to hold, 
the labor is doubled by using a five-place instead of a four-place table, and 
is increased one-half by using a six instead of a five place table. Hence, as 
there is no sensible gain from using an excess of places, it is obviously very 
important to employ a table of the smallest admissible number of places. But, 
on the other hand, the use of too few places must be guarded against. As an 
instance of a somewhat dangerous practice may be cited the use of four-place 



LOGARITHMS. 



XXVII 



tables in o. i per cent work. This is a not infrequent practice, most common 
perhaps in chemical computation, and, of course, arising from the exceeding 
convenience of cards containing four-place logarithms. It will be shown at 
page xliv, that four places are not sufficient for o. i per cent direct computations, 
and the error if four-place logarithms are used is sensibly the same. The com- 
putation error may easily rise to o. 2 or o. 3 per cent with four-place tables even 
in ordinary computations. 

To Find the Logarithms of a Number. 

Rule. — Regard the number Q as separated into two factors 
q X 10*", where q begins in the units' place (see " Notation by 
Powers of 10," page xv). Find in the tables the logarithm of q. 
This will be the mantissa of the' desired logarithm. Prefix to this 
the characteristic or index ± n. 

A few examples will sufficiently elucidate the process. 

Iizample. — Desired the logarithm to four decimal places of the number 306. 

Write the number, or, better, merely consider it as if factored in the form 
3.06-10^. In the four-place table, on the line 3.0 and in the column headed 
6 will be found .4857, which is log 3.06. Obviously, log lo^ = 2. Therefore 
log 306. = log 3.06 + log io2 = .4857 -f 2, which is usually written 2.4857. 

Further Bsamples. Interpolation. — Desired to four decimal places the 
logarithm of 306.2. 

This will lie between the logs of 306 and 307 (and approximately 0.2 of the 
way), and as the table is not carried out further we must interpolate. 

Difference = .0014, usually written 
14. 

The desired mantissa will be 0.2 of 
the way from .4857 to .4871. Hence 
we must add to the former number 
0.2-14. = 3- 

The interpolation may always be made by subtracting and multi- 
plying as in this example, but to save the labor, logarithm tables 
are usually provided with marginal interpolation tables, by the aid 
of which interpolations may easily be made mentally. 

Thus in taking out log 3.062 we find log 3.06 = .4857- By inspection, differ- 
ence = 14. In the interpolation table headed 14, line 2, stands 3, which is there- 
fore the desired 0.2 of 14. Therefore log 3.062 = .4860. This operation could, 
of course, be carried otit mentally. 

The present tables are arranged so that the interpolation tables stand oppo- 
site, or nearly so, to the logarithms to which they correspond. This not only 
gives them a convenient location but enables the computer usually to avoid 
even the mental subtraction of the successive logarithms to find the difference, 
since this will, of course, be that at the bead of the nearest difference table. 
Usually also the error introduced by using an interpolation table slightly too 
large or too small is negligible. 



For 3.07 we find .4871 

For 3.06 we find -4857 

Interpolation, 0:2 of 14 = 2.8 = 3 

. •. log 3.062 = .4860 
log 306.2 = log 3.062 -I- log 10^ = 2.4860 



xxviii LOGARITHMS. 

Interpolation becomes more laborious and more liable to error, 
if conducted mentally, in proportion as the difference is large. It 
is therefore greatest in the first quarter of any logarithm table. But 
it happens that in physical, chemical, and engineering computations 
there very often enter correction or reduction factors and other 
terms of the form (i + a) or i/(i — a), where a is a decimal frac- 
tion rarely as large as o.i. The frequency of such terms calls for 
a disproportionately large number of the more laborious interpola- 
tions. To avoid this labor and increased chance of error, the excel- 
lent practice has been adopted in some five-place tables of inserting 
two pages giving the logarithms from i.o to i.i, by steps only one- 
tenth as large as in the rest of the tables, thus doing away with 
all interpolation in this most used and most troublesome portion 
of the table, without adding seriously to its bulk. Such a table 
has here been prefixed to the five-place table. In the four-place 
table the same result has been accomplished here, in a manner which 
is perhaps novel, by the insertion of ten additional lines at the head 
of the table. The Vega seven-place tables unfortunately lack this 
feature. 

To find Logarithm of a Decimal Fraction. — The procedure is 
precisely the same as for a whole number. Note that the log- 
arithm of a decimal fraction is always negative, and, conversely, 
that a negative characteristic always denotes a decimal fraction. 

Example. — Desired the log of o.oo 306 2. 

log 0.00 306 2 = log 3.o62TO~' 

log 3.07 = .4871 

3.06 = .4857 

o. 2 of difference = 3 

. •. log 3.062 = .4860 

log 10-3 = - 3. 

. •. log 0.00 306 2 = .4857 — 3. 

This is written eitiier 3.4857 or 7.4857 — 10. The latter form is ohtalned by 
adding 10 to the characteristic and appending — 10 to the whole. The num- 
bers thus appended must in many instances, but not in all, be followed up in 
the computation in order to correctly locate the decimal point at the ■ close. 
This is, however, usually very little trouble. The second method is the very 
general practice find is based on the assertion that when several logarithms are 
to be added, it is more convenient to have all the characteristics positive. The 
author is, however, of the opinion that this conventional method serves almost 
no useful purpose, and that it is better and less troublesome in every way to 
retain the negative characteristic. It is almost as easy to add a column of num- 
bers in which some are negative, and are therefore subtracted when they are 



Difference = 14. 



LOGAKITHMS. XXIX 

reached instead of being added, as to add a column in which all are positive. 
And if the negative characteristic is retained, all care and writing of — lo or 
its multiples is avoided. Moreover, the logarithm is complete in itself and 
shows at once that it is the log of a decimal fraction. These remarks apply not 
only to the logarithms of ordinary numbers but to the logarithms of the trigono- 
metric functions. 

Example of addition where some of the characteristics are negative. Places 
beyond the first of the mantissa are not added in this illustration. 

2.4036 Beginning at the bottom of the columns we have 

T.2168 5 + 3 = 8 + 2 = 10 + 4 = 14, write 4 ; 

1-3462 carry 1 + 2 = 1+ I = + 1=1 + 2 = 1. 

^•^"^ Of course the figures printed in heavy type are the only ones pro- 

1.4 nounced or mentally enforced. 

Ezamples of some cases where the use of the notation by powers of 10 
enable one to dispense with the characteristic in portions of computations are 
given incidentally at page xxi. 

Grouping by Fives in the Tables. — Throughout the entire set 
of tables, it will be observed, the columns and lines are arranged in 
groups of five. This not only aids the eye to readily follow any 
desired line or column, but enables the computer with a little prac- 
tice to enter a desired column or line without glancing at the num- 
ber at the top of the column or side of the line, and similarly to read 
off the nvmiber of column cr line without glancing off to the marginal 
figure. Thus the middle column in the second group is 7, the last 
in the first group is 4, and so on. The practice of working by 
observed position rather than by the marginal number should be 
pursued. It reduces fatigue and tends to prevent mistakes. 

Indexing at Corners of Tables. — The bold type at the outer cor- 
ners of the tables shows the contents of the page and its opposite. 
Thus on the sixth page of the five-place log table, the larger black 
figure 2 shows that these two pages contain the logs of all the num- 
bers beginning with 2. The smaller black figures, 30, 47, in the 
margin show that the mantissae on the two pages extend from 30 to 
47 (first two figures). The indexing will be found a material help 
in rapidly running the page corners under the finger torfind either a 
desired table or a desired figure in a table. 

Decimal Point in Logarithm Tables. — It is the almost universal 
custom to omit the decimal point entirely from logarithm tables. 
This tends toward compactness, but is by no means essential on that 
score. The omission causes no serious inconvenience after slight 



ixx' LOGARITHMS. 

practice. On the other hand, the retention of the point renders the 
table complete and strictly self -consistent ; that is, any mantissa in 
the table is then the completely expressed logarithm of the cor- 
responding tabular number. This fact tends materially toward clear 
comprehension on the part of the beginner or occasional computer. 
The table is also then perfectly assimilated to the "Notaltion by 
Powers of lo," page xv, which affords not only the clearest basis of 
explanation of mantissa and characteristic, but by far the easiest 
method of obtaining and following the characteristic in computations. 
The points do not add necessarily to the bulk of the tables and are 
no hindrance to their use by computers accustomed to other rules 
than those here given respecting the characteristic. They are 
therefore retained in these tables. 

Antilogarithm = Number Corresponding. — Given the logarithm to 
find the "number corresponding," i.e. the number of which this is 
the logarithm. This number is also called the " antilogarithm." 

From the Log Tables. — 

Ilzample. — To And the antilog of 2.4857, inspect the body of the four-place 
log table to find the mantissa .4857 (or the next smaller than this if the exact 
value does not appear). It will be found to lie on line 3.0 and in column 6, and 
Is, hence, the log of 3.06. The characteristic 2 is the log of lo^. 
.•. antilog 2.4857 = 3.o6-io2 or 306. 

Example. — To find the antilog of 2.4860. In the table this mantissa does 
not appear exactly, but the next smaller is .4857, whose antilog is 3.06, and the 
difference from this to the next larger is 14. 

.4860 - .4857 = 3, and t'j = .2, 
.". antilog .4860 is -j^ or 0.2 of the way from 3.06 to 3.07, 
.•. antilog .4860 = 3.062, 
also antilog 2. := lo^, 

.'. antilog 2.4860 = 3.062-10^ or 306.2. 

The interpolation can be mentally made by the marginal interpolation tables. 
The tabular difference is 14, and it is desired to know what decimal fraction the 
difference 3 is of this. Looking down the column under 14 for the number 
nearest to 3 it is found to be 3 and to stand opposite to 2. .'. 3 is 0.2 of 14, and 
antilog .4860 = 3.062. 

It will be noticed that in the four-place and five-place log tables 
those mantissse have been printed in heavier type in which the first 
figure changes from one digit to the next. This serves as a guide 
to the eye in looking for any desired mantissa whose antilog is sought. 

From Tables of Antilogarithms. — Some computers prefer to employ 
a special table for antilogarithms instead of working backward in 
the ordinary logarithm table as in the preceding example. Whether 



LOGARITHMS. XXxi 

the small saving in time effected by this means is an equivalent 
to the slight additional mental effort incident to the employment in 
alternation of tables differently arranged, may be questioned. Where 
many antilogs are to be taken out in succession, the gain is, however, 
sensible. 

Ksample. — To find by the table of Antilogarithms the antilog 2.4857. 

Taking the mantissa first, we find on line .48, column 5, the next smaller 
number 3.055 with a tabular difference of 7. Hence, antilog .485 = 3.055. But 
antilog .4857 will be 0.7 of the way from that of .485 to .486. The amount to be 
added then for interpolation is o. 7 x 7 = 5. 

antilog .4857 = 3.055 + 0.005 = 3-o6o, 
antilog 2.4857 = 3.060-102 =306.0. 

Interpolation tables are provided whose method of use is sufficiently 
obvious. 

Cologarithms = Logarithms of Reciprocals. — The colog of any 
number jp is the- logarithm of the reciprocal of the number. 

It is therefore log i — logp = o — logj9. 

In substitution in ,a formula such as x = "•"•" ^ which is the product of 

several numbers (one or more) divided by the product of several others (one or 
more), the direct process would be to take the sum of the logarithms of the 
factors in the numerator, and to subtract from this the sum of the logs of the 
factors in the denominator. 

Example. — log a = i. 2543 log = 3. 8642 

l0g6 =3.8766 log(i = 5.2I2I 

loge = 2.1345 



Numerator sum = 1. 1309 Denominator sum = 1.2108 

Subtract denominator sum= 1.2108 



Difference = 3.9201 
a; = number corresponding = 8.320'io"', or 0.008320. 

This process may be simplified by employing the colog. It then becomes 

logos =1.2543 

log 6 = 3.8766 
colog c =4.1358 
colog (^ = 4.7879 
colog e =3.8655 

Sum = 3.9201 
X = number corresponding = 8.320-10-3 

This process is thus reduced to the simple addition of a series of numbers. 

The colog may be easily taken out of the usual log table by 
merely looking out the logarithm and subtracting mentally from o. 



XXXll 



LOGARITHMS. 



Example. — Desired the colog of 306.2. Log 306.2 = 2.4860, colog = 3.5140. 

The use of cologs either with or without a table effects but small saving, 
except in case a series of substitutions in a given formula are to be made, so 
that a number of cologs may be looked out in immediate succession. 

Table of Cologs. — The table of four-place cologarithms is arranged 
similarly in every respect to the table of four-place logarithms, and 
the cologs are taken from it just as logs from their table; noting, 
however, that the cologs in the table have the characteristic i, and 
that the differences are subtractive. 



Example. — Desired the colog of 306.2. 

In line 3.0, column 6, colog 3.06 
0.2 X (— 14) by difference table 

.-. colog 3.062 
colog 10* 



Separate into 3.o62-io2. 



= 1-5143 
= -3 



Difference from 
colog 3. 07 is — 14. 



= 1^.5140 
= 2. 



colog 306.2 = colog 3.062- + colog io2 = 3-5140 



Example. — Desired the colog of 0.00 713 6. 

In line 7.1, column 3, is colog 7.13 = 

0.6 (— 6) by difference table z 

colog 7. 1 36 = 

colog 10^3 = 

colog 0.00 713 6 = colog 7.136 -f- colog IO~3= 



1. 1469 

7.1465 

+ 3- 
2.1465 



Difference is — 6. 



Habit in Reading off Numbers or Logarithms. — Time can be econ- 
omized, strain on the attention reduced, and liability to mistake 
lessened by an easily acquired habit of grouping and emphasizing 
the figures in reading off the numbers, the mantissa, and the num- 
bers corresponding (antilogs) in using tables. 

A good method of reading is as follows : 

In reading a number or antilog pay no regard to the decimal 
point. Emphasize the first figure; pause, read second and third 
figures; pause, read remaining figures in groups of three. Thus: 
Desired the log of 30.62 047 2. Read this as 3 06 204 72, i.e. three 
. . . naught six . . . two naught four . . . seven two. 

In reading off a mantissa use no emphasis, but group the first 
two figures together, and the subsequent figures by threes. Thus, the 
mantissa .4869 would be read as 48 69, i.e. forty-eight . . . sixty- 
nine; and .48601 as 48 601, i.e. forty-eight . . . six naught one. 

In taking from the log table the numbers corresponding (antilog) 
to .48601, it would be read 306 2, three . . . naught six . . . two, 
precisely as under the above rule for reading a number. 



LOGARITHMS. XXXlii 

These rules apply equally well to four or five place tables. In 
tables of six or seven places the first three figures of the mantissa 
are grouped together in printing, and are therefore more conveniently 
read oS together. Also, it is more convenient to read off six-place 
numbers and antilogs with three figures instead of two in the second 
group ; thus, 7 814 62. 

The difierence between a number and a mantissa thus read off, whether 
audibly or mentally, almost precludes the possibility of mistaking one for the 
other, so that less strict attention will be required to avoid entering the number 
column of a table with a mantissa, or vice versa. It also avoids the mental or 
verbal employment of words of instruction. Thus, if a computer reads off 7 82 
he knows, or his assistant using the log tables knows, as soon as the first figure 
has been read that the logarithm of the number is desired. Conversely, if 
he reads off 43 857, the first two figures alone show that the quantity is a man- 
tissa, and that the antilog is required. Thus, no words of instruction need be 
used throughout an entire computation, and yet no possibility of error need 
enter. 

It is also to be noted that this grouping is consistent with the symmetrical 
grouping advocated at an earlier page ; adapts itself perfectly to the employment 
of the notation by powers of 10 ; and coincides with the most convenient group- 
ing in the flve-place tables. 

Powers and Roots by Logarithms. — If a = lo" (so that m = log a) 
then (a)" = (lo")" and log («■")= n log a. Here both m and n may 
be either positive or negative, and either an integer or a fraction. 

Hence, to raise any number, whole or fractional, to any power, 
integral or fractional, and positive or negative, multiply the logarithm 
of the number by the exponent of the power of the number. 

Since a root is a fractional power, i.e. y« = ^s, the above rule 
includes the extraction of a root. In the case of decimal fractions 
the fact that the characteristic is negative while the mantissa is pos- 
itive, must be regarded. The correct result will be assured if the 
mantissa and characteristic are separately multiplied (or divided) 
by the exponent, and the latter result then subtracted from the for- 
mer, as will be further shown. Certain special procedures will also 
be described. 

The one exception to this is where the index of the power is a single figure 
and positive. This case takes care of Itself without special attention. 

It may seem that the use of the negative characteristic thus increases labor 
over the usual notation explained at the foot of page xxviii. Inspection of 
examples will show that the only case in which it does so is where the index of 
the power or root is other than a single digit. 



XXxiv LOGAKITHMS. 

Powers and Roots of Numbers greater than Unity. — 
Examples. 



Desired the cube of 


47.16,1.6.(47.16)'. 


log 47.16 
Multiply by 


= 1.673s 
3 


log a; 


= 5.0205 .•. a; = i.048'ic 


Desired \/47i.6. 




log47i.6 
Divide by 


= 2.6735 
5 


log a; 


= 0.5347 .-. a; =3.425, 



Ezample. — Desired the 2.416 power of 65 830. 

log 65 830. = 4.8184 

Multiply by 2.416 

96368 
I 92736 

4818 

2892 

.-.log (65 830)2-«« =11.6413 

(65 83o)2«6 _ 4.378 • 10" = 437800000000. 

To avoid the direct multiplication of the logarithm by the index of power, 
when this contains several figures, their logs may be taken. Thus 

(S pi. logs.) (4 pi. logs.) 

log log 65 830 = log 4.8184 = 0.68291 0.6829 

log 2.416 = 0.38310 0.3831 

loglog (65 83o)2-^i«= 1.06601 1.0660 

antilog 1.06 601 = log (65 83o)"'6 = 11.641 11.64 

.-. (65830)2116= 4.376-1011 4.365-1011. 

When the log of a log is thus taken, a table giving at least one 
place more in the mantissa should be used than would be otherwise 
needed in order to protect the last place of figures in the result, as 
is shown in the above example, which is by no means an extreme 
case. If the quantity is a decimal fraction, the negative character- 
istic must be separately treated and the result subtracted from the 
result obtained with the mantissa. 

Powers of Decimal Fractions. — Here it is necessary to notice that 
the logarithm of a decimal fraction is composed of two parts, one 
positive the other negative. Thus, 

logo.o6 831 = 2.8345 or 8.8345-10., 
according to the notation chosen. In the first form (the one here 
recommended) the logarithm consists of a negative characteristic 



LOGARITHMS. XXXV 

and a positive mantissa ; in the second form it consists of a positive 
logarithm (both characteristic and mantissa) followed by a negative 
characteristic. 

No new procedure is necessary, but it is essential to be consistent 
as to sign of each part, and to remember that both parts must be 
subjected to any operation performed upon either. Whenever the 
exponent has more than one figure, the negative characteristic and 
the positive mantissa should be separately multiplied by the index, 
and the former subtracted from the latter. 

Example. — Desired (0.004716)*. 

logo.oo 4716 = 3.6735 or 7.6735-10 

Multiply by 4 4 



10.6940 or 30.6940 — 40, or 0.6940 — 10 

.-. (0.004716)* = number corresponding, or antilog, 
= 4.943.io~w or 0.00 000 000 049 43. 

Example. — Desired (p.oo 4^16)*-^^. 



logo.00 4716 =3.6735 or 
4-32 


7.6735 - 10 
4-32 4-32 


26940 
20215 
1348 


306940- 43.2 
230205 
15348 


2.9095 
— 12.96 


33-14953 
-43-2 


17.9495 


1T.949S 


(o.oo47i6)'*-32 = 2.962'io~ii. 





Roots of Decimal Fractions. — To extract the root of a decimal 
fraction, divide separately the negative characteristic and the posi- 
tive mantissa by the index of the desired root. Subtract the former 
quotient from the latter. The difference will be the logarithm of 
the desired root. In other words, treat the characteristic separate 
from the logarithm. This procedure is always the safest to adopt 
when there is any doubt in the mind of the computer. 

Example. — Desired Vo.oe 831. 

log 0.06 831 = 2.8345 

Dividing characteristic by 3 = — 0.6667 

Dividing mantissa by 3=0. 2782 



.. log V0.06831 = 1.6115 
v'o.oe 831 = number corresponding — 4.088-10-1. 



XXXvi SQUARES AND SQUARE BOOTS. 

A method easily understood by inspection is this : Add to the neg- 
ative characteristic of the logarithm a number equal to mr where r 
is the index of the desired root and m is a whole number large 
enough to make mr larger than or equal to the characteristic, in 
other words, large enough to extinguish the negative sign of the 
characteristic. Write this number with- a negative sign after the 
logarithm. Then divide the whole by the index r. The quotient 
will be the logarithm of the desired root. Usually m = i, i.e. 



Example. — Desired \/o.o6 83i- 
log 0.06 831 = 2.8345, adding and subtracting 3 gives 1-8345 ~ 3 

dividing by 3 gives 0.6115 — i = 1.6115 
.*. ■\/o.o5 831 ="number corresponding = 4.o88'io-i or 0.4088. 



SQUARES AND SQUARE ROOTS. 

It is common to give separate tables of squares and- of square 
roots by which, respectively, the square or square root of any num- 
ber may be taken out. Inspection, however, shows that the four- 
place table of square roots must occupy four pages, the first two 
containing the numbers from i.o to 10., the second two from 10. to 
100. ; the corresponding roots ranging from i. to 10. ; while a table of 
squares would occupy but two pages, containing numbers from i.o to 
10., the squares ranging from i. to 100. Hence the tabular diifer- 
ences in the table of square roots will be much smaller than in the 
table of squares. For this reason the table of squares may advan- 
tageously be dispensed with, and squares be taken out when desired 
from the table of square roots, as the numbers corresponding (anti- 
logarithms) are taken out of a table of logarithms. The smallness 
of the differences makes interpolation easier and more rapid than in 
a table of squares, and this more than offsets the slight disadvan- 
tage of the reverse process of interpolation. The tables at page 30 
give square roots to four places. As in the logarithm tables, the 
numbers in the body of the table are printed in stronger type wher- ' 
ever the second figure changes, in order to assist the eye in the 
reverse use of the table. 

In the table of square roots the insertion of the extra section giving i.oo to 
1. 10 to form places direct to avoid interpolation is not called for as in logarithms 
because the interpolation is very easy, and because the squares of terms of the 
form (i -^ a) are not frequently required. 



SQUARES AND SQUARE ROOTS. XXXVll 

To find the square of any number, factor the number as described 
in the "Notation by Powers of lo," page xv. Enter the body of 
the table with the first factor, and find the corresponding marginal 
reading and column heading which will be the first three figures 
of the square. Interpolate for the fourth figure. Square the 
factor lo*", which makes it io±^" Multiply together these two 
squares. 

Example. — Desired the square of 34 850. 

34 850.2 =(3.485-101)2= 3.4852- I02X^. 

In the table 3.479 (the next smallest value to 3485) stands in line 12, col- 
umn I. The tabular difference is 14, of which our difference ( = 3.485 — 3.479=6) 
is y\, or 0.4 (as may be seen at once from the interpolation table headed 14). 

Hence 34850.2 = 12. 14-108= 121 400000. 

Example. — Desired (0.00049 83)2. • 

(0.0004981)2 = 4.98i2-io-*x2 = 24.83-io"8 = 0.00000024 83. 

To find the square root of any number, factor it as in the " Nota- 
tion by Powers of 10," page xv, except that n must be no-sv an even 
number, -while the first factor must have either one or two digits pre- 
ceding the decimal point, whichever may be necessary, in order to 
permit n to be made even. 

Find, then, from the table, interpolating if necessary, the square 
root of the finst factor, and multiply this by 10-"^^, the square root 
of the second factor ; the product is the desired square root. 

Example. — Desired the square root of 347.6. 
(347.6)i=(3.476-io2)i 

In the table, line 3.4, column 7, gives (3.47)2 = 1.863 
Interpolations by table gives 0.6 difference = i 

(3.476)^=7^86^ 

.. (347.6)^= 1. 864-ioi= 18.64. 

Example. — Desired ■v/875 200. 

V87.S2-I0* = 9-355''o^ = 93S-S- 

Example. — Desired (0.05643) ?. 

(5.643-10-2)7 = 2.376-10-1 = 0.2376. 

Example. — Desired 0.00 006 784. 

67.84- io"« = 8.236-10-3 = 0.008236. 



XXXviii KECIPKOCALS. 



RECIPROCALS. 

This table, page 34, contains tlie reciprocals to four places of 
numbers from i. to 9.99, and by iaterpolation from i. to 9.999. The 
reciprocal to four places of any number from zero to infinity is, 
of course, one of these tabular values multiplied by a suitable 
power of 10. In this table, the reciprocals of numbers from i. to 
1. 1 00 are given directly in the first ten lines to avoid interpolation 
in finding the reciprocals of terms of the form (i + a), where a is 

a small decimal fraction. 

* 

N.B. — The approximation 1/(1 +a) = (i —a) approx., is frequently em- 
ployed in computations to avoid dividing by (i + a). This useful approxima- 
tion, however, must he used with caution. The error from its employment is 
a^ ; that is, if a = o.i, the error from multiplying by (i — a) instead of divid- 
ing by (i 4- a) is 0.12 = 0.01, or i per cent; if = 0.03, the error is 0.03^ 
= 0.0009, 01' nearly o. i per cent, and so on. 

To find the reciprocal of a number, factor it by the " Notation by 
Powers of 10," page xv.' Find the reciprocal of the first factor by 
the table and multiply this by the second factor 10=*=" with the alge- 
braic sign of its exponent reversed, i.e. by lo'f". 

Example. — Desired 1/4486,1.6: (4486) ~i. 

(4486.)-! =(4.486- io8)-i =(4.486.)-i-io-3. 
In table, line 4.4, col. 8, (4.48)-! = 0.2232 

By difference table, 0.6 difference = — 3 

Reciprocal of 4.486 = 0.2229 

(4486.)-! = (4.486)~i-io-8 = 0.2229-IO"' = 0.0002229. 

Example. — Desired i/o.oo 327 5. 

(0.00 327 S)-i =(3.275)-i-(io-8)-i = o.3os4-io» = 305.4. 



NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 

It is frequently convenient to have a table giving rough values 
of the natural trigonometric functions for use in preliminary, check, 
or approximate computations. The four-place tables of the above 
four functions cover all ordinary needs. Interpolation can be 
made in these tables to o°.oi by the interpolation tables, or to i' 
by mental computation, since the step between successive columns is 
o°.i, or 6'. 



TEIGONOMETEIC FUNCTIONS. XXXIX 

Example. — Desired the natural sine of 37°. 75. 

On line 37° under heading °.'] is 0.61 15. To interpolate for the remaining 
figure we must add o.j of the tabular difference, which, by inspection, is 14. 
0.5 X 14 = 7, as can be seen at once in the interpolation table 14 opposite the 
line 37°. 

.-. sin 37°.75 =0.6115 + .0007 =0.6122. 

Example. — Desired the natural cotangent of 72°. 28. 

The cosines and cotangents read upward in the right-hand degree column. 

Line 72°, column °.2, gives 0.321 1 

0.8 of difference (19), to be subtracted, — 15 

.•. natural cotangent 72°.28 =0.3196 

All of the tables are used similarly. 



LOGARITHMS OF SINES, COSINES, TANGENTS, AND COTANGENTS. 

Two sets of tables are given, four-place and five-place, respec- 
tively. The four-place tables read to tenths and hundredths of 
degrees, and are convenient for general rough work and with instru- 
ments reading to tenths of a degree. The five-place tables give 
values to minutes direct. They are adapted to a very large part of 
the angular measurements ordinarily made in experimental work. 
Interpolation to half or quarter minutes is easily made in them 
mentally. A computer working closer than that would naturally 
employ the Vega, or other conveniently arranged six- or seven-place 
tables, dropping unnecessary places. 

For the reasons stated under the discussion of logarithm tables, 
page xxviii, the negative characteristic is here retained instead of the 
more customary 9, with the appended — 10. 

The use of the tables needs little explanation. The four-place 
tables are used precisely as the tables of the natural functions. In 
the five-place tables, for less than 45", read the tables downwards, 
using the minute column at the left hand. Tor angles greater than 
45°, read the tables upwards, using the minute column at the right 
hand of the table, and take care to employ the column-headings 
given at the foot of the table. 

Examples. — 

Log sin 31° 22' = 7.71 643. 
Log tan 54° 46' =0.15 loi. 



xl SLIDE WIliE RATIOS. 



SLIDE WIRE RATIOS. 

The increasing use of the slide wire in electrical measurements, 
notably in connection with physical chemistry, renders the table of 
values of 



a decided convenience. The work to which the slide wire is thus 
for the most part employed demands but four-place tables. 

To use the table, let a be the reading in millimetres on the slide wire, 
which is supposed to be one metre long with millimetre subdivision, 
or of any other length with division into thousandths. Find the 
first two figures of a in the first column, and run out on this line to 
the column headed with the third figure. The number there found 
will give the value of the above fraction, which is the ratio of the 
length of one section of the wire to the other. If a contains a fourth 
figure, that is, if read to tenths of a millimetre, interpolation must be 
made in the usual manner. 

Example. — The slider reads 415.6 millimetres. 

In line 41 under 5 is found 0.7094. Interpolating by adding 0.6 of the 
tabular diflerence 29, gives 

X = 0.7094 + 0.6 X 29 

= 0.7094+ .0017 =0.7111. 



DEFINITIONS AND EXPLANATIONS 

UNDERLYING THE COMPUTATION RULES. 

The following statements call for the attention of those only who find 
unfamllar terms in the foregoing rules. 

A Digit is any one of the ten characters i, 2, 3, 4, 5, 6, 7, 8, 9, o. 

A Significant Figure is any digit used to denote or signify the amount of the 
quantity in the place in which it stands. Thus zero may be a significant figure 
when it is written not merely to locate the decimal point, but to indicate that 
the quantity in the place in which it stands is known to be nearer to zero than 
to any other digit. 

For example, if a distance has been measured to the nearest fiftieth of an 
inch and found to be 205.46 inches, all five of the figures, including the zero, are 
significant. And similarly if the measurement had shown the distance to be 
nearer to 205.40 than to 205.41 or to 205.39 the filial zero would be also signifi- 
cant, and should invariably be retained, since its presence serves the most 
useful purpose of showing that this place of figures had been measured as well as 
the rest. If in such a case the quantity had been written 205.4 instead of 205.40, 
the inference would be drawn either that the hundredths of an inch had not been 
measured, or that the person who wrote the number was ignorant or careless of 
the proper numerical usage. Failure to follow this simple rule is a common 
source of annoyance and uncertainty. 

A zero when used merely to enable the decimal point to be retained is of 
course not a significant figure in the above sense. E.g. If the distance were 
measured as 286. centimetres within ± i centimetre, it might be retained as 
2.86 metres, or 0.00 286 liilometres, or 2860. millimetres. In this case neither of 
the zeros would be significant. 

From this last example it is obvious that when the first zero, or zeros, pre- 
cede the decimal point, they fail to indicate intrinsically whether or not they are 
significant. E.g. The number 2860. millimetres or 28 600. millimetres, standing 
apart from explanatory context, would afford no clew to whether the last place, 
or the last two places, had been measured. In writing such a number, therefore, 
whenever it is desirable to convey this information some statement of the reliabil- 
ity of the quantity must be appended. This is usually done by writing after the 
number ± a where a is a number, of two significant figures only, representing 
the estimated measure of the accuracy or unreliability of the result. (See later. ) 

Places of Figures are the places in which figures stand in the number as 
actually written. Places of significant figures are those in which significant 
figures stand. These two terms being merely, although not actually, identical, 
are often used interchangeably. 

xli 



xlii DEFINITIONS AND EXPLANATIONS. 

Example. — 426018. has six places of figures and of significant figures 
3 479 100. has seven places of figures, hut whether its numher of places of signifi- 
cant figures is more than five is indeterminate. 0.02 7680 has seven places of 
figures with five places of significant figures. 2.76 8o-io-''' has five places of 
figures and five significant figures. 

Places of Decimals. — These, of course, are the places following the decimal 
point as the numher happens to be vrritten. E.g. 0.027680 has six decimal 
places. 2.7680-10'^ has four decimal places, although its magnitude is the 
same as that of the preceding quantity. 

From the foregoing six examples it will be seen that the number of deci- 
mal places and the number of places of significant figures have no necessary- 
mutual relation. 

Accuracy ; Reliability. — To know the exact accuracy of a given quantity 
it would be necessary, of course, to know the true value of that quantity. In 
the case of a few mathematical constants (such as ir = 3.14 159 265 •.•) the true 
value is known, at least far beyond ordinary requirements. But in case of all 
measurements the same is obviously not true, for if the true value were known, 
measurements would be unnecessary. Some approximate expression of the accu- 
racy of a measured result can, however, usually be obtained, and is necessary. 
Sometimes this is afforded by a knowledge of the instrument used and the degree 
of care employed. Thus, suppose the distance of about 3 feet 6^^ inches 
= 3-5039 feet between two marks to have been carefully once measured with a 
good foot-rule, it could safely be assumed from our previous knowledge of such 
vyork that if a series of these measurements were undertaken, the results would 
vary from their mean by less than ± 3V of an inch ; also, that the error in the 
rule itself, and from other unavoidable or unavoided sources, would not on the 
average materially increase the error of measurement. Then ± -J^ inch 
= ± 0.0026 ft. would be taken as the estimated measure of accuracy of the result. 
Instead of expressing the accuracy in units, e.g. inches or feet, it is usually 
more convenient or intelligible to express it as a fraction ; or, better still, in per- 
centage. Thus, the foregoing will be 

, 0.0026 , .74 / , 0.0026\ , i 

± = ± 0.00 074, I.e. — '-^ — , or looi ± ^ 1 = ± 0.074 per cent. 

3-S looooo \ 3.5 j 

If in this example the reliability had not been estimated at ^^ inch, but if 
several measurements had been made, and these had been found upon inspec- 
tion to deviate from their mean by about j^j of an inch, then, other things being 
the same, the measure of accuracy of any single measurement taken without 
knowledge of the others would be regarded as ± -r}^ inch, or ± 0.074 per cent. 

Mean ; Average ; Deviation Measure. — When the result is the arithmetical 
mean or average of several separate measurements of the same quantity, its 
reliability or accuracy is taken to be in proportion to the square root of the 
numher of such observations. Thus, in the last preceding example, if there had 
been m = 9 single observations made, the measure of accuracy of the mean of 
these vf ould be 

± -i- -7- Vre = ± -5 — \z = db 4: = ± o.oio inch, or ± 0.025 per cent. 
32 32 ^9 96 

The differences of the single observations taken under like conditions from 
their mean will be called deviations, and their numerical average (i.e. their sum 



DEI'INITIONS AND EXPLANATIONS. xliii 

omitting their algebraic sign, divided by their number) will be called the average 
deviation of the single observation from the mean, and will be denoted by ad. 
This quantity divided by the square root of n would be called the average devia- 
tion of the mean, and will be denoted by AD. It is the averagS amount by 
which any one such mean would be found to deviate from the average of a 
number of such means all taken under like conditions. The term deviation 
measure will be used in referring to either of these quantities. 

Briefly, then, the meaning of the terms may be summarized as follows : By 
the statement that the accuracy or reliability of a result is of a quoted amount 
is meant that the deviation measure of this result is estimated or found to be of 
this stated amount, and that so far as this can be discovered by inspection, no 
other sources of error exist which aflect the result by an amount sensible as 
compared with this.* 

It is essential to bear ir» mind in connection with all quantities that are the 
result of measurement that no absolute numerical expression of the accuracy 
or error is possible ; that any expression which is given is usually merely a 
deviation measure; that is, an approximate average value of the effect of 
the variable parts of the errors, accompanied by an assurance, expressed or 
implied, that a study of the discoverable sources of error of the process has 
been made, and that these have been corrected for or found to be negligible 
compared to the deviation measure. 

In specifying the accuracy desired in the result, it must be understood that 
merely the converse of this is meant ; or, at least, that only the converse is possi- 
ble of attainment. 

Rules for Significant Figures. — The rules are so framed that, barring mis- 
takes, the greatest possible computation error entering into the result of any 
ordinary computation (e.g. one involving a total of not much exceeding 20 
component numbers, steps, or operations, where a rejection error may occur) 
shall not be sensible compared with the errors of the measurements or data, or 
shall not sensibly afEect the accuracy of the result. They are, therefore, safe 
rules in the worst possible cases. But in order to be so they are necessarily 
more than sufficiently stringent for some classes of comparatively rough work, 
where the infrequent undetected entrance of a computation error two to four 
times as large as the experimental error would be permissible. For such work 
one less place of figures may be used, but when the rules are thus relaxed the 
possible consequence should be borne in mind, and special scrutiny applied to 
the various stages of the computation, special attention being directed to quan- 
tities beginning with i or 2. 

Rejection Error. — Whenever it becomes necessary to throw off places of 
figures, a " rejection error " may enter into the result ; e.g. suppose that for any 
reason the last two figures are to be rejected in 

24 375 291, 
making it 24 375 300. 

The rejection error in the new form is evidently -|- 9. 

In rejection, the last figure retained is always to be increased by i when the 
rejected figure next it is 5 or over, but remains unchanged if that figure is less 
than 5. 

* For a more extended discussion of this and allied subjects see the author's 
" Precision of Measurements." 



xliv DEFINITIONS AND EXPLANATIONS. 

Thus, calling the last place retained the rth place, the limits of the rejection 
error entering into that place are + 0.5 and — 0.5, and as all amounts between 
these limits are equally likely to occur, the average rejection error in the long 
run will be 6.25 in the rth place. 

Law and Amount of Accumulated Rejection Error. — Let the last place 
of significant figures retained in a number, e.g. the fourth, fifth, etc., be called 
the rth place. Then the error entering from rejection of figures beyond the rth 
will be at most ± 5 in the (»■ + i) place, and any error between these limits, 
+ 5 and — 5 in the (?• + i) place, will be equally likely to occur in any given 
case, and therefore will be of equal frequency in the long run. The average 
rejection error in any considerable number of rejections will therefore be ± 2.5 
in the (r + i) place. If then in direct processes of multiplication, division, evolu- 
tion, or involution (separate or combined) each factor, product, and quotient in 
the operation be carried out to the same number r of places, what will be the accu- 
mulated fractional rejection error if n such rejections are made during the entire 
operation ? This error we shall call for brevity the computation error, or 
simply the rejection error. This question might easily be answered in a form 
giving the average accumulated error, but we are at present concerned chiefly 
with the maximum possible error, since our object is to frame rules which will 
reduce the worst possible computation error to negligible dimensions. The 
maximum computation error would arise when every single rejection error was 
5 in the (r 4- i) place, and all had the same Eign ; and this would be the greatest 
fraction of the final result when that result and all the factors (and therefore all 
the intermediate products, quotients, etc.) began with i and had o in the other 
places, i.e. were each ic— ' with the decimal point wherever it might happen to 
stand. If there were n of these factors, products, quotients, etc., at which 
rejections were made, the maximum possible computation error in the result 
■would, therefore, be re x 5 in the (r+ i) place, and the fractional error would 
be 5 re/ic. Since this maximum error would be exceedingly rare unless n were 
very small, and any approaching one-half of it would be very rare, we may 
properly assume that our rules will be sufficiently stringent if we allow this 
maximum error to have the same magnitude as the desired accuracy in the 
result as expressed on the basis of the precision measure, or deviation measure, 
explained at page xlii. To determine most simply what number r of places this 
limitation would call for in the processes under consideration, let us take specific 
cases. Suppose the work is desired to possess an accuracy of i per cent. Then 
5 re/ic must be equal to or less than i per cent; i.e. 5 re^i/ioo. Hence, we 
have to solve 





5 re _ I 
IC 100 




By inspection, if 


'• = 3. Sn= 10, 


.-. re = 2, 


if 


r = 4, 5 n = 100, 


.'. re = 20. 



But obviously n will almost never be as small as 2, and rarely as large as 20, 
lying with greatest frequency between 5 and 10 and averaging below 7. This 
will give a maximum error of 0.0035 o"" j P^r cent with re = 7, r = 4, which would 
be insignificant, and rising to i per cent only when re = 20. Hence, in work of 
multiplication, division, etc., where i per cent accuracy, or a little better, is 
desired (remembering that by this we mean only a deviation measure of i per 
cent) r = 4 will be an entirely safe but not an excessive value ; that is, the reten- 



DEFINITIONS AND EXPLANATIONS. xlv 

tion of four significant figures throughout will insure entirely sufBoient freedom 
from computation error in every case when the number of rejections is less than 
the very unusual total of about 20, and fewer places would not be warranted. 
Similarly five places will suffice for worlt to o.i per cent, i.e. better than i per 
cent, but not much better than o.i per cent; six places for work of o.oi per 
cent, and so on. 

As to relaxation of the rules in special cases, it is evident that but little can 
be safely done. The maximum error with n = 7, »■ = 4, will be 0.35 per cent, 
and an error of \ of this, say, of 0.1 per cent, will not be of very uncommon 
occurrence. In fact, with only two rejections the error might be 0.1 per cent. 
Hence, four places cannot be considered as safe to o. i per cent, even for short 
computations. Four places might properly enough be used in short computa- 
tions up to J per cent. Similar statements of course hold for the other rules. 

When logarithms are used for multiplication, division, etc., tables should be 
used giving the mantissa to the same number of places of figures as required by 
the. foregoing rules for direct multiplication and division. Hence, one would 
employ four-place tables for i per cent work, five-place tables for o.i per cent 
work, and so on. The numbers, antilogs, and mantissse should all be carried 
out to the same number of places. This conforms to the customary and only 
convenient practice. 

This rule is based primarily on the fact, next to be shown, that under them 
the maximum computation error in the use of logarithms arises chiefly from the 
rejection in the numbers and antilogs themselves and not sensibly from the 
rejected places in the logarithms. In logarithms, a change of the rth figure by i 
produces the same fractional error in the antilog whatever its value, viz. 2.4/10'', 
as may be seen easily by inspection of tables. Hence, as the maximum rejection 
error in the tabular value of a logarithm is 5 in the (»• 4- i) place of the man- 
tissse, which may be doubled by the process of interpolating, the maximum 
fractional error in a result due to the maximum rejection error in any mantissa 
is 2.4/10''. But if only r places are retained in the number or antilog, the 
maximum error in it due to rejection of its further- places is s/ic, compared 
to which 2.4/10' is negligible. Hence, as the number of rejections from num- 
ber and antilog together is usually about the same as from mantissse, the accumu- 
lated error will be due almost wholly to the rejections from the numbers and 
antilogs. And of these rejections there will ordinarily be about as many if the 
computation is carried out by logarithms as if by direct multiplication or 
division. The above rule is thus justified. 

In addition or subtraction the maximum rejection error will be obviously 
5 TO in the (r -f i) place. Under the above stated rule that the weakest quantity 
shall be carried to four significant figures (or two uncertain places) for i per 
cent work, the smallest value of the deviation measure or uncertainty of the 
result win be 10 in the rth place, which is 100 in the (r -f i) place. Hence, 

5™^ 100. .-. n = 20. 

The maximum accumulated error would then attain the size of the smallest 
deviation measure, i.e. the worst possible case would occur, only when the num- 
ber of rejections was as great as twenty. Hence, the rule of four places for 1 
per cent work, five for 0.1 per cent work, and so on, as before given, is sufficient. 



TABLES 



LOGS. 4 PL. 



FOUR PLACE LOGARITHMS. 



No. 





I 


2 


3 


4 


5 


6 


7 8 


9 


INTERPOLATION 
TABLES. 


1.00 


.0000 


.0004 


.0009 


.0013 


.0017 


.0022 


.0026 


.0030 .0035 


.0039 


38 36 34 32 


.01 


0043 


0048 


0052 


0056 


0060 


0065 


0069 


0073 0077 


0082 


4 4 3 3 


.02 


0086 


0090 


0095 


0099 


0103 


0107 


OIII 


0116 0120 


0124 


8776 


•03 


0128 


0133 


0137 


0141 


0145 


0149 


0154 


0158 0162 


0166 


II II 10 10 


.04 


0170 


0175 


0179 


0183 


0187 


0191 


0195 


0199 0204 


0208 


15 14 14 13 


1.05 


.0212 


.0216 


.0220 


.0224 


.0228 


•0233 


.0237 


.0241 .0245 


.0249 


19 18 17 16 


.06 


0253 


0257 


0261 


0265 


0269 


0273 


0278 


0282 0286 


0290 


23 22 20 19 


.07 


0294 


0298 


0302 


0306 


0316 


0314 


0318 


0322 0326 


0330 


27 25 24 22 


.08 


0334 


0338 


0342 


0346 


0350 
0390 \ 


°354 


035S 


0362, 0366 


0370 


30 29 27 26 


.09 


0374 


0378 


0382 


0386 


0394 


039^ 


0402 0406 


0410 


34 32 31 29 
30 28 26 24 


1.0 


.0000 


.0043 


.0086 


.0128 


.0170 


.0212 


.0253 


.0294 .0334 


■0374 


.1 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 0719 


0755 


3332 


.2 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038. 1072 


1 106 


6655 


•3 


"39 


"73 


1206 


1239 


1271 


i3°3 


1335 


1367 1399 


1430 


9887 


•4 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 1703 


1732 


12 II 10 10 


'5 


.1761 


.1790 


.1818 


.1847 


.1875 


.1903 


•1931 


•1959 .i987«2014 


15 14 13 12 


.6 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 2253 


2279 


18 17 16 14 


•7 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 2504 


2529 


21 20 18 17 


.8 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 2742 


2765 


24 22 21 19 


■9 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 2967 


2989 


27 25 23 22 


2.0 


.3010 


•3°32 


■3054 


■3075 


.3096 


.3118 


•3139 


.3160 .3181 


.3201 


22 20 18 16 


.1 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 3385 


3404 


2222 


.2 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 3579 


3598 


4 4 4 3 


•3 


3617 


3636 


3655 


3674 


3692 


37" 


3729 


3747 3766 


3784 


7655 


•4 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 3945 


3962 


9876 


2-S 


•3979 


■3997 


.4014 


.4031 


.4048 


.4065 


.4082 


.4099 .4116 


•4133 


II 10 9 8 


.6 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 4281 


4298 


13 12 II 10 


•7 


43H 


4330 


4346 


4362 


4378 


4393 


4409 


4425 4440 


4456 


15 14 13 II 


.8 


4472 


4487 


4502 


45>8 


4533 


4548 


4564 


4579 4594 


4609 


18 16 14 13 


•9 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 4742 


4757 


20 18 16 14 


3.0 


•4771 


.4786 


.4800 


.4814 


.4829 


•4843 


.4857 


.4871 .4886 


.4900 


15 14 13 12 


.1 


4914 


4928 


4942 


4955 


4969 


4983 


49^7 


6011 5024 


5038- 


2 I I I 


.2 


5051 


5065 


5079 


5092 


5105 


5"9 


5132 


5145 5'S9 


5172 


3 3^3 2 


•3 


5185 


5198 


5211 


5224 


5237 


5250 


5263 


5276 5289 


5302 


5 4 4.. 4 


•4 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 5416 


5428 


6 6 5 5 


35 


•5441 


•5453 


.5465 


•5478 


•5490 


.5502 


•5515 


■5527 -5539 


•5551 


8776 


.6 


5563 


5575 


5587 


5599 


561 1 


5623 


5635 


5647 5658 


5670 


9887 


■7 


5682 


5694 


5705 


5717 


5729 


S740 


5752 


5763 5775 


5786 


II 10 9 8 


.8 


5798 


5809 


5821 


5832 


5843 


5855 


5866 


5877 5888 


5899 


12 II 10 10 


•9 


S9II 


5922 


5933 


5944 


5955 


5966 


5977 


5988 5999 


6010 


14 13 12 II 


4.0 


.6021 


.6031 


.6042 


•6053 


.6064 


.6075 


.6085 


.6096 .6107 


.6117 


11 10 9 8 


.1 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 6212 


6222 


I I I I 


.2 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 6314 


6325 


2222 


■3 


633s 


6345 


6355 


6365 


6375 


6385 


6395 


6405 6415 


6425 


"3332 


•4 


6435 


6444 


6454 


6464 


6474 


6484 


6493 


6503 6513 


6522 


4 4 4 3 


4-S 


•6532 


.6542 


.6551 


.6561 


•6571 


.6580 


.6590 


.6599 .6609 


.6618 


6554 


.6 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 6702 


6712 


7655 


■7 


6721 

^^12 
6902 


6730 


6739 


6749 


6758 


6767 


6776 


6785 6794 


6803 


8766 


.8 


6821 


6830 


6839 


6848 


6857 


6866 


6875 6884 


6893 


9876 


•9 


691 1 


6920 


6928 


6937 


6946 


6955 


6964 6972 


6981 


lo 9 8 7 


tnc 


i^'Dl'V' 










['>\ 











LOGS 



FOUR PLACE LOGARITHMS. 



4 PL. LOGS. 



No. 





I 


2 3 


4 


5 


6 


7 


8 


9 


nterpolationI 

TABLES. 1 


5.0 


.6990 


.6998 


.7007 .7016 


.7024 


•7033 


.7042 


.7050 


.7059 


.7067 


9 8 


7 


.1 


7076 


7084 


7093 7101 


7H0 


7118 


7126 


7135 


7143 


7152 


I I 


I 


.2 


7160 


7168 


7177 7185 


7193 


7202 


7210 


7218 


7226 


7235 


2 2 


I 


•3 


7243 


7251 


7259 7267 


727s 


7284 


7292 


7300 


7308 


7316 


3 2 


^ 


•4 


7324 


7332 


7340 7348 


7356 


7364 


7372 


7380 


7388 


7396 


4 3 


3 


5-5 


.7404 


.7412 


.7419 .7427 


•7435 


•7443 


•7451 


•7459 


.7466 


•7474 


5 4 


4 


.6 


7482 


7490 


7497 7505 


7513 


7520 


7528 


7536 


7543- 


7551 


5 5 


4 


•7 


. 7559 


7566 


7574 7582 


7589 


7597 


7604 


7612 


7619 


7627 


6 6 


5 


.8 


7634 


7642 


7649 7657 


7664 


7672 


7679 


7686 


7694 


7701 


7 6 


6 


•9 


7709 


7716 


7723 7731 - 


^7738 


7745 


7752 


7760 


7767 


7774 


8 7 


6 


6.0 


.7782 


.7789 


.7796 .7803 


.7810 


.7818 


.7825 


•7832 


•7839 


.7846 


7 


6 


.1 


7853 


7860 


7868 7875 


7882 


7889 


7896 


7903 


7910 


7917 


I 


I 


.2 


7924 


7931 


7938 7945 


7952 


7959 


7966 


7973 


7980 


7987 


I 


I 


•3 


7993 


8000 


8007 8014 


8021 


8028 


8035 


8041 


8048 


8055 


2 


2 


•4 


8062 


8069 


8075 8082 


8089 


8096 


8102 


8109 


8116 


8122 


3 


2 


6.5 


.8129 


.8136 


.8142 .8149 


.8156 


.8162 


.8169 


.8176 


.8182 


.8189 


4 


3 


.6 


8195 


8202 


8209 8215 


8222 


8228 


8235 


8241 


8248 


8254 


4 


4 


•7- 


8261 


8267 


8274 8280 


8287 


8293 


8299 


8306 


8312 


8319 


5 


4 


.8 


8325 


8331 


8338 8344 


8351 


8357 


8363 


8370 


8376 


8382 


6 


5 


■9 


8388 


8395 


8401 8407 


8414 


8420 


8426 


8432 


8439 


8445 


6 


5 


7.0 


.8451 


•8457 


.8463 .8470 


.8476 


.8482 


.8488 


.8494 


.8500 


.8506 


6 


5 


.1 


8513 


8519 


8525 8531 


8537 


8543 


8549 


8555 


8561 


8567 


I 


I 


.2 


8573 


8579 


8585 8591 


8597 


8603 


8609 


8615 


8621 


8627 


I 


I 


•3 


8633 


8639 


8645 8651 


8657 


8663 


8669 


8675 


8681 


8686 


2 


2 


•4 


8692 


8698 


8704 8710 


8716 


8722 


8727 


8733 


8739 


8745 


2 


2 


7-5 


.8751 


.8756 


.8762 .8768 


.8774 


.8779 


.8785 


.8791 


.8797 


.8802 


3 


3 


.6 


8808 


8814 


8820 8825 


8831 


8837 


8842 


8848 


8854 


8859 


4 


3 


.7 


8865 


8871 


8876 8882 


8887 


8893 


8899 


8904 


8910 


8915 


4 


4 


.8 


8921 


8927 


8932 8938 


8943 


8949 


8954 


8960 


8965 


8971 


5 


4 


•9 


.8976 


8982 


8987 8993 


8998 


9004 


9009 


9015 


9020 


9025 


5 


5 


8.0 


.9031 


.9036 


.9042 .9947 


•9053 


.9058 


.9063 


.9069 


.9074 


.9079 


6 


6 


.1 


9085 


9090 


9096 9101 


9106 


9112 


9117 


9122 


9128 


9133 


I 


I 


.2 


9138 


9143 


9149 9154 


9159 


9165 


9170 


9175 


9180 


9186 


I 


I 


•3 


9191 


9196 


9201 9206 


9212 


9217 


9222 


9227 


9232 


9238 


2 


2 


•4 


9243 


9248 


9253 9258 


9263 


9269 


9274 


9279 


9284 


9289 


2 


2 


8.5 


.9294 


.9299 


.9304 .9309 


•9315 


.9320 


•9325 


•9330 


•9335 


•9340 


3 


3 


.6 


9345 


9350 


9355 9360 


9365 


9370 


9375 


9380 


9385 


9390 


4 


3 


.7 


9395 


9400 


9405 9410 


9415 


9420 


9425 


9430 


9435 


9440 


4 


4 


.8 


9445 


9450 


9455 9460 


9465 


9469 


9474 


9479 


9484 


9489 


5 


4 


•9 


9494 


9499 


9504 9509 


9513 


9518 


9523 


9528 


9533 


9538 


5 


5 


9.0 


•9542 


■9547 


■9552 .9557 


.9562 


.9566 


•9571 


■9576 


.9581 


.9586 


5 


4 


.1 


959° 


9595 


9600 9605 


9609 


9614 


9619 


9624 


9628 


9633 


I 





.2 


9638 


9643 


9647 9652 


9657 


9661 


9666 


9671 


9675 


9680 


I 


I 


•3 


9685 


9689 


9694 9699 


9703 


9708 


9713 


9717 


9722 


9727 


2 


I 


■4 


9731 


9736 


9741 9745 


9750 


9754 


9759 


9763 


9768 


9773 


2 


2 


9-5 


•9777 


.9782 


.9786 .9791 


•9795 


.98CXJ 


.9805 


.9809 


.9814 


.9818 


3 


2 


.6 


9823 


9827 


9832 9836 


9841 


9845 


9850 


9854 


9859 


9863 


3 


2 


•7 


9868 


9872 


9877 9881 


9886 


9890 


9894 


9899 


9903 


9908 


4 


3 


.8 


9912 


9917 


9921 9926 


9930 


9934 


9939 


9943 


9948 


9952 


4 


3 


•9 


9956 


9961 


9965 9969 


9974 


9978 


9983 


9987 


9991 


9996 


5 


4 



(3) 



4 PL. LOGS. 



FOUR PLACE ANTILOCARITHMS. 

ANTILOGS. 4 PL. 









T H 


USA 


N D T H 8. 






INTERPOLA. 1 


Mant. 





I 


2 3 


4 


567 


8 


9 


TABLES. 1 


.00 


1. 000 


1.002 


1.005 '007 


1.009 


1.012 I.OI4 1.016 


1.019 


1.021 


2 


3 


.01 


1.023 


1.026 


1.028 1.030 


1-033 


1.03s '-038 1.040 


1.042 


1.045 








.02 


1.047 


1.050 


1.052 1.054 


1.057 


1.059 1.062 1.064 


1.067 


1.069 





I 


■03 


1.072 


1.074 


1.076 1.079 


1.081 


1.084 1086 1.089 


1.091 


1.094 




1 


.04 


1.096 


1.099 


1.102 1.104 


1.107 


1.109 I-H2 1.114 


1.117 


1.119 




1 


. -05 


1. 122 


1.125 


1.127 I-I30 


1.132 


1.135 I-I38 1.140 


I -143 


1. 146 




2 


.06 


1. 148 


1.151 


1.153 1.156 


1-159 


1. 161 1.164 1-167 


1.169 


1.172 




2 


.07 


1. 175 


1.178 


1. 180 1.183 


1.186 


1.189 1-191 I-I94 


1.197 


1.199 




2 


.08 


1.202 


1.205 


1.208 1.21 1 


1.213 


1.216 1.219 1-222 


1.225 


1.227 


2 


2 


•09 


1.230 


1-233 


1.236 1.239 


1.242 


1.245 1-247 1^250 


1-253 


1.256 


2 


3 


.lO 


1.259 


1.262 


1.265 '-268 


1.271 


1.274 1.276 1.279 


1.282 


1.285 


3 


4 


.11 


1.288 


1. 291 


1.294 1-297 


1.300 


1.303 1.306 1.309 


1.312 


I-3I5 








.12 


I.318 


1.321 


1.324 1.327 


1-33° 


1-334 1-337 1-340 


1-343 


1.346 


1 


1 


•13 


1^349 


1-352 


1-355 1-358 


1.361 


1.365 1.368 1.371 


1-374 


1-377 


1 


1 


.14 


1.380 


1.384 


1.387 1.390 


1^393 


1.396 1.400 1.403 


1.406 


1.409 


1 


2 


•15 


1-413 


1.416 


1.419 1.422 


1.426 


1.429 1.432 1-435 


1-439 


1.442 


2 


2 


.16 


1.445 


1.449 


1.452 1.455 


'•459 


1.462 1.466 1.469 


1.472 


1.476 


2 


2 


•17 


1-479 


1-483 


1.486 1.489 


1-493 


1.496 1.500 1.503 


1.507 


1.510 


2 


3 


.18 


1.514 


r.517 


1.521 1.524 


1.528 


1-531 1-535 1-538 


1.542 


1-545 


2 


3 


.19 


1-549 


1-552 


1.556 1.560 


1-563 


1.567 1.570 1.574 


1.578 


1.581 


3 


4 


.20 


1.585 


1-589 


1.592 1.596 


1.600 


1.603 1-607 i.6n 


1.614 


1.618 


3 4 


6 


.21 


1.622 


1.626 


1.629 I -633 


1-637 


1.641 1.644 I-648 


1.652 


1.656 





1 


.22 


1.660 


1.663 


1.667 1-671 


1675 


1.679 1.683 1-687 


1.690 


1.694 


1 1 


I 


•23 


1.698 


1.702 


1.706 1.710 


1.714 


1.718 1.722 1.726 


1-730 


1-734 


1 1 


■ 2 


.24 


1-738 


1.742 


1.746 1.750 


1-754 


1.758 1.762 1.766 


1.770 


1-774 


I 2 


2 


•25 


1.778 


1.782 


1.786 1.791 


1-795 


1.799 1.803 1-807 


1.811 


1.816 


2 2 


3 


.26 


1.820 


1.824 


1.828 1.832 


1-837 


1.841 1.845 I-849 


1.854 


1.858 


2 2 


3 


•27 


1.862 


1.866 


1.871 1.875 


1.879 


1.884 1-888 1.892 


1.897 


1.901 


2 3 


4 


.28 


1-905 


1.910 


1-914 i-9>9 


1.923 


1.928 1.932 1.936 


1.941 


1-945 


2 3 


4 


•29 


1.950 


1.954 


1-959 1-963 


1.968 


1.972 1.977 1-982 


1.986 


1.991 


3- 4 


5 


■30 


1-995 


2.000 


2.004 2.009 


2.014 


2.018 2.023 2.028 


2.032 


2.037 


4 5 


6 


•31 


2.042 


2.046 


2.051 2.056 


2.061 


2.065 2.070 2.075 


2.080 


2.084 


I 


I 


•32 


2.089 


2.094 


2.099 2.104 


2.109 


2.113 2.118 2.123 


2.128 


2.133 


I 1 


1 


•33 


2.138 


2.143 


2.148 2.153 


2.158 


2.163 2.168 2.173 


2.178 


2.183 


1 2 


2 


•34 


2.188 


2.193 


2.198 2.203 


2.208 


2.213 2.218 2.223 


2.228 


2-234 


2 2 


2 


•35 


2.239 


2.244 


2.249 2.254 


2.259 


2.265 2.270 2.275 


2.280 


2.286 


2 3 


3 


•36 


2.291 


2.296 


2.301 2.307 


2.312 


2.317 2.323 2.328 


2-333 


2.339 


2 3 


4 


•37 


2-344 


2.350 


2.355 2.360 


2.366 


2-371 2.377 2.382 


2-388 


2.393 


3 4 


4 


•38 


2-399 


2.404 


2.410 2.415 


2.421 


2.427 2.432 2.438 


2-443 


2.449 


3 4 


5 


•39 


2.455 


2.460 


2.466 2.472 


2.477 


2.483 2.489 2.495 


2.500 


2.506 


4 5 


5 


.40 


2.512 


2.518 


2.523 2.529 


2-535 


2.541 2.547 2.553 


2-559 


2.564 


5 6 ' 


r 8 


.41 


2.570 


2.576 


2.582 2.588 


2.594 


2.600 2.606 2.612 


2.618 


2.624 


1 I 


1 


.42 


2.630 


2.636 


2.642 2.649 


2-655 


2.661 2.667 2.673 


2.679 


2.685 


1 I 


2 


•43 


2.692 


2.698 


2.704 2.710 


2.716 


2.723 2.729 2.735 


2.742 


2.748 


2 2 : 


! 2 


■44 


2-754 


2.761 


2.767 2.773 


2.780 


2.786 2.793 2.799 


2.805 


2.812 


2 2 ; 


i 3 


•45 


2.818 


2.825 


2.831 2.838 


2.844 


2.851 2.858 2.864 


2.871 


2.877 


3 3 4 4 


.46 


2.884 


2.891 


2.897 2.904 


2.911 


2.917 2.924 2.931 


2.938 


2-944 


3 4 4 5 


•47 


2.951 


2.958 


2.965 2.972 


2.979 


2.985 2.992 2.999 


3.006 


3-013 


4 4 ; 


6 


.48 


3.020 


3.027 


3.034 3.041 


3.048 


3.055 3.062 3:069 


3.076 


3-083 


4566 


•49 


3.090 


3-097 


3.105 3.112 


3-"9 


3.126 3.133 3.141 


3.148 


3-IS5 


556-7 



ANTILOGS. 4 PL. 



(4) 



FOUR PLACE ANTILOCARITHMS. 

4 PL. ANTILOGS. 











THOUSANDTHS. 








INTERPOLA. 


Mant. 





I 


2 


3 4 


5 


6 


7 


8 


9 


TABLES. 


.50 


3.162 


3-170 


3-177 


3.184 3.192 


3-199 


3.206 


3-214 


3-221 


3.228 


7 8 9 


•SI 


3^236 


3-243 


3-251 


3.258 3.266 


3-273 


3.281 


3.289 


3.296 


3-304 


I 1 I 


.52 


3^3 J I 


3-319 


3-327 


3-334 3-342 


3-350 


3-357 


3-365 


3-373 


3-381 


122 


•S3 


3.388 


3-396 


3-404 


3.412 3.420 


3-428 


3-436 


3-443 


3-451 


3-459 


223 


•S4 


3-467 


3475 


3-483 


3.491 3.499 


3.508 


3-516 


3-524 


3-532 


3-540 


3 3 4 


•SS 


3-548 


3-556 


3-565 


3-573 3.581 


3-589 


3-597 


3.606 


3.614 


3.622 


4 4 5 


.56 


3-631 


3-639 


3-648 


3.656 3.664 


3-673 


3-681 


3.690 


3-698 


3-707 


4 5 5 


•57 


3-715 


3-724 


3-733 


3-741 3-750 


3-758 


3-767 


3-776 


3-784 


3-793 


566 


.58 


3.802 


3-8 1 1 


3-819 


3.828 3.837 


3.846 


3-855 


3-864 


3-873 


3.882 


667 


•S9 


3.890 


3-899 


3.908 


3-917 3-926 


3.936 


3-945 


3-954 


3-963 


3972 


6 7 8 


.60 


3-981 


3-99° 


3-999 


4.009 4.018 


4.027 


4.036 


4.046 


4.055 


4.064 


9 10 11 12 


.6i 


4.074 


4.083 


4-093 


4.102 4.111 


4.121 


4. J 30 


4.140 


4-150 


4-159 


I I I 1 


.62 


4.169 


4.178 


4.188 


4.198 4.207 


4.217 


4.227 


4.236 


4.246 


4.256 


2222 


•63 


4.266 


4.276 


4-285 


4.295 4.305 


4.315 


4.325 


4-335 


4-345 


4-355 


3 3 3 4 


.64 


4-365 


4-375 


4.385 


4.395 4.406 


4.416 


4.426 


4-436 


4-446 


4-457 


4 4 4 5 


•6S 


4.467 


4-477 


4.487 


4.498 4.508 


4.519 


4-529 


4-539 


4-550 


4.560 


5 5 6 6 


.66 


4-571 


4.581 


4.592 


4.603 4.613 


4.624 


4-634 


4-645 


4.656 


4.667 


5677 


.67 


4-677 


4.688 


4-699 


4.710 4.721 


4-732 


4-742 


4-753 


4.764 


4-775 


6788 


.68 


4.786 


4-797 


4.808 


4,819 4.831 


4.842 


4-853 


4.864 


4-875 


4.887 


7 8 9 10 


.69 


4.898 


4.909 


4.920 


4.932 4.943 


4.955 


4.966 


4-977 


4-989 


5.000 


8 9 10 11 


.70 


5.012 


5-023 


5-035 


5.047 5.058 


5.070 


5.082 


5 -093 


5.105 


5.117 


12 13 14 15 


•71 


S.129 


5-140 


5-152 


5.164 5.176 


5.188 


5.200 


5.212 


5.224 


5-236 


1112 


•72 


5-248 


5.260 


5.272 


5.284 5.297 


5 -309 


S-321 


5-333 


5.346 


5-358 


2333 


•73 


5-370 


5-383 


5-395 


5.408 5.420 


5-433 


S-445 


5-458 


5.470 


5-483 


4 4 4 5 


•74 


5-495 


5.508 


5-521 


,5.534 5-546 


5-559 


5-57« 


S-585 


5.598 


5.610 


5566 


•75 


5-623 


5^636 


5-649 


5-662 5.675 


5.689 


5-702 


S-715 


5-728 


5-741 


6778 


.76 


5-754 


5.768 


5.781 


5.794 5.808 


5-821 


5-834 


5-848 


5.861 


5-875 


7889 


•77 


5.888 


5.902 


5.916 


5-929 S-943 


5-957 


S-970 


5-984 


5-998 


6.0J2 


8 9 10 11 


.78 


6.026 


6.039 


6.053 


6.067 6.081 


6.095 


6.109 


6.124 


6.138 


6.152 


10 10 11 12 


•79 


6.166 


6.180 


6.194 


6.209 6.223 


6.237 


6.252 


6.266 


6.281 


6-295 


11 12 13 14 


.80 


6.310 


6.324 


6-339 


6.353 6.368 


6-383 


6-397 


6.412 


6.427 


6.442 


16 17 18 19 


.81 


6.457 


6.471 


6.486 


6.501 6.516 


6-531 


6.546 


6.561 


6.577 


6.592 


2222 


.82 


6.607 


6.622 


6.637 


6.653 6.668 


6.683 


6.699 


6.714 


6.730 


6-745 


3 3 4 4 


•83 


6.761 


6.776 


6.792 


6.808 6.823 


6.839 


6.855 


6.871 


6,887 


6.902 


5556 


.84 


6.918 


6.934 


6.950 


6.966 6.982 


6.998 


7.015 


7031 


7-047 


7.063 


6778 


•8S 


7.079 


7.096 


7. 112 


7.129 7.145 


7.161 


7.178 


7- "94 


7.211 


7.228 


8 9 9 10 


.86 


7.244 


7.261 


7.278 


7-295 7-3" 


7-328 


7-345 


7.362 


7.379 


7.396 


10 1011 11 


.87 


7-413 


7-430 


7-447 


7.464 7.482 


7-499 


7-516 


7-534 


7-551 


7.568 


II 121313 


.88 


7.586 


7-603 


7.621 


7.638 7.646 


7.674 


7.691 


7-709 


7.727 


7-745 


13141415 


.89 


7.762 


7.780 


7.798 


7.816 7.834 


7-852 


7.870 


7.889 


7-907 


7.925 


14 15 16 17 


.90 


7-943 


7.962 


7.980 


7.998 8.017 


8.035 


8.054 


8.072 


8.091 


8.110 


20 2122 23 


.91 


8.128 


8.147 


8.166 


8.185 8.204 


8.222 


8.241 


8.260 


8.279 


8.299 


2222 


.92 


8.318 


8-337 


8.356 


8-375 8.395 


8.414 


8-433 


8-453 


,8.472 


8,492 


4 4 4 5 


■93 


8.5 1 1 


8-531 


8.551 


8.570 8.590 


8.610 


8.630 


8.650 


8.670 


8.690 


6677 


•94 


8.710 


8.730 


8.750 


8.770 8.790 


8.810 


8.831 


8.851 


8.872 


8.892 


8899 


•95 


8.913 


8.933 


8.954 


8.974 8.995 


9.016 


9.036 


9.057 


9.078 


9.099 


10 11 11 12 


.96 


9.120 


9.141 


9.162 


9.183 9.204 


9.226 


9.247 


9.268 


9.290 


9-3^ J^ 


12131314 


•97 


9-333 


9-354 


9-376 


9-397 9-419 


9.441 


9.462 


9,484 


9.506 


9.528 


1415 15 16 


.98 


9-550 


9-572 


9-594 


9.616 9.638 


9.661 


9-683 


9-705 


9.727 


9-750 


16 17 18 18 


•99 


9.772 


9-795 


9.817 


9.840 9.863 


9.886 


9.908 


9-931 


9-954 


9-977 


18 19 20 20 



(5) 



4 PL. ANTILOGS. 



FOUR PLACE COLOGARITHMS. 



COLOGS. 4 PL. 



NOTE THE CHARACTERISTIC 1. 



No. 


01234 


56789 


INTERPOLATION 
TABLES. 


1,00 


r.9996 T.9991 T.9987 ^-9983 


7.9978 7.9974 7.9970 7.9965 7.9961 




.01 


1-9957 9953 9948 9944 9940 


9935 9931 9927 9923 99i8 




.02 


9914 9910 9905 9901 9897 


9893 9887 9884 9880 9876 




•03 


9S72 9867 9863 9859 9855 


9851 9846 9842 9838 9834 




.04 


9830 9825 9821 9817 9813 


9809 9805 9801 9796 9792 


Use the flnl ten 
liDes to avoid in- 


1.05 


T.9788 T.9784 T.9780 7.9776 T.9772 


7.9767 7.9763 7.9759 7.9755 7.9751 


terpolating 

from liOOO 


.06 


9747 9743 9739 9735 9731 


9727 9722 9718 9714 9710 


to 1,100. 


.07 


9706 9702 -9698 9694 9690 


9686 9682 9678 9674 9670 




.08 


9666 9662 9658 9654 9650 


9646 9642 9638 9634 9630 




.09 


9626 9622 9618 9614 9610 


9606 9602 9598 9594 9590 


-44-40-36-32 


1.0 


0.0000 T.9957 7.9914 7.9872 7.9830 


7.9788 7.9747 7.9706 7.9666 7.9626 


.1 


7.9586 9547 9508 9469 9431 


9393 9355 93i8 9281 9245 


4 4 4 3 


.2 


9208 9172 9136 9101 9066 


9031 8996 8962 8928 8894 


9876 


•3 


8861 8827 8794 8761 8729 


8697 8665 8633 8601 8570 


13 12 II 10 


•4 


8539 8508 8477 8447 8416 


8386 8356 8327 8297 8268 


18 16 14 13 


i-S 


7.8239 7.8210 7.8182 7.8153 7.8125 


7.8097 7.8069 7.8041 7.8013 7.7986 


22 20 18 16 


.6 


7959 7932 7905 7878 7852 


7825 7799 7773 7747 772i 


26 24 22 19 


•7 


7696 7670 7645 7620 7595 


7570 7545 7520 7496 7471 


31 28 25 22 


.8 


7447 7423 7399 7375 7352 


7328 7305 7282 7258 7235 


35 32 29 26 


•9 


7212 7190 7167 7144 7122 


7100 7077 7055 7033 7011 


40 36 32 29 


2.0 


7.6990 7.6968 7.6946 7.6925 7.6904 


7.6882 7.6861 7.6840 7.6819 7.6799 


-28-26-24-22 


.1 


6778 6757 6737 6716 6696 


6676 6655 6635 6615 6596 


3322 


.2 


6576 6556 6536 6517 6497 


6478 6459 6440 6421 6402 


6554 


•3 


6383 6364 6345 6326 6308 


6289 6271 6253 6234 6216 


8877 


•4 


6198 6180 6162 6144 6126 


6108 6091 6073 6055 6038 


II 10 10 9 


2-S 


7.6021 7.60037.59857.59697.5952 


7.5935 7.5918 7.5901 7.58847.5867 


14 13 12 II 


.6 


5850 5834 5817 5800 5784 


5768 5751 5735 5719 5702 


17 16 14 13 


•7 


5686 5670 5654 5638 5622 


5607 5591 5575 5560 5544 


20 18 17 15 


.8 


5528 5513 5498 5482 5467 


5452 5436 5421 5406 5391 


22 21 19 18 


■9 


5376 5361 5346 5331 5317 


5302 5287 5272 5258 5243 


25 23 22 20 


3.0 


7.5229 7.5214 7.5200 7.5186 7.5 1 71 


7.5157 7.5143 7.5129 7.5114 7.5100 


-18-16-14r-12 


.1 


5086 5072 5058 5045 5031 


5017 5003 4989 4976 4962 


2 2 I I 


.2 


4948 4935 4921 4908 4895 


4881 4868 4855 4841 4828 


4332 


•3 


4815 4802 4789 4776 4763 


475° 4737 4724 47" 4698 


5 5 4 4 


•4 


4685 4672 4660 4647 4634 


4622 4609 4597 4584 4572 


7665 


3-5 


MS59 '■.4547 7-4535 M522 7.4510 


7.4498 7.4486 7.4473 7.4461 7.4449 


9876 


.6 


4437 4425 4413 4401 4389 


4377 4365 4353 4342 4330 


II 10 8 7 


.7 


4318 4306 4295 4283 4271 


4260 4248 4237 4225 4214 


13 II 10 8 


.8 


4202 4191 4179 4168 4157 


4145 4134 4123 4112 4101 


14 13 II 10 


•9 


4089 4078 4067 4056 4045 


4034 4023 4012 4001 3990 


16 14 13 II 


4.0 


7.3979 7-3969 7-3958 7.3947 7.3936 


7.3925 7.3915 7.3904 7.3893 7.3883 


-11-10 -9 -8 


.1 


3872 3862 3851 3840 3830 


3820 3809 3799 3788 3778 


I I I I 


.2 


3768 3757 3747 3737 3726 


3716 3706 3696 3686 3675 


2222 


.3 


3665 3655 3645 3635 3625 


3615 3605 3595 3585 3575 


3332 


•4 


3565 3556 3546 3536 3526 


3516 3507 3497 3487 3478 


4 4 4 3 


4-S 


7.3468 7.3458 7.3449 7.3439 7.3429 


7.3420 7.3410 7.3401 7.3391 7.3382 


6 5 5 4 


.6 


3372 3363 3354 3344 3335 


3325 3316 3307 3298 3288 


7655 


.7 


3279 3270 3261 3251 3242 


3233 3224 3215 3206 3197 


8766 


.8 


3188 3179 3170 3161 3152 


3143 3134 3125 3116 3107 


9876 


' 


3098 3089 3080 3071 3063 


3054 3045 3036 3028 3019 


10 9 8 7 



COLOGS. 4 PL. 



(6) 



FOUR PLACE COLOCARITHMS. 

4 PL. COLOGS. 



No. 


o I 2 3 4 


56789 


interpolation! 

TABLES. 1 


5.0 


7.3010 T.3002 T.2993 T.2984 T.2976 


7.2967 7.2958 7.2950 7.2941 7.2933 


-9 


-8 -7 


.1 


2924 2916 2907 2899 2890 


2882 2874 2865 2857 2848 


1 


1 1 


.2 


2840 2832 2823 2815 2807 


2798 2790 2782 2774 2765 


2 


2 I 


■3 


2757 2749 2741 2733 2725 


2716 2708 2700 2692 2684 


3 


2 2 


•4 


2676 2668 2660 2652 2644 


2636 2628 2620 2612 2604 


4 


3 3 


5-5 


T.2595 T.2588 T.2581 T.2573 T.2565 


7.25577.25497.2541 7.25347.2526 


S 


4 4 


.6 


2518 2510 2503 2495 2487 


2480 2472 2464 2457 2449 


S 


5 4 


■7 


2441 2434 2426 2418 241 1 


2403 2396 2388 2381 2373 


6 


6 5 


.8 


2366 2358 2351 2343 2336 


2328 2321 2314 2306 2299 


7 


6 6 


•9 


2291 2284 2277 2269 2262 


2255 2248 2240 2233 2226 


8 


7 6 


6.0 


T.22I8T.22U 7.2204 T.2197 1.2190 


7.2182 7.2175 7.2168 7.2161 7.2154 


-7 


-6 


.1 


2147 2140 2132 2125 2I18 


2111 2104 2097 2090 2083 


I 


1 


.2 


2076 2069 2062 2055 2048 


2041 2034 2027 2020 2013 


1 


1 


■3 


2007 2000 1993 1986 1979 


1972 1965 1959 1952 1945 


2 


2 


•4 


1938 I93I 1925 I918 I9II 


1904 1898 1891 1884 1878 


3 


2 


6.5 


7.1871 7.1864 7.1858 7.185I 7.1844 


7.18387.18317.18247.18187.1811 


4 


3 


.6 


1805 1798 1791 1785 1778 


1772 1765 1759 1752 1746 


4 


4 


■7 


1739 1733 1726 1720 1713 


1707 1701 1694 1688 1681 


5 


4 


.8 


1675 1669 1662 1656 1649 


1643 1637 '^3° '624 1618 


6 


5 


■9 


1612 1605 1599 1593 1586 


1580 1574 1568 1561 1555 


6 


5 


7.0 


7.1549 7.1543 7.1537 7.1530 7.1524 


7.15187.15127.15067.15007.1494 


-6 


-5 


.1 


1487 I481 1475 1469 1463 


1457 1451 1445 1439 1433 


I 


I 


.2 


1427 I42I 1415 1409 1403 


1397 1391 1385 1379 1373 


I 


I 


•3 


1367 1361 1355 1349 1343 


1337 1331 1325 '319 1314 


2 


2 


•4 


1308 1302 1296 1290 1284 


1278 1273 1267 1261 1255 


2 


2 


7-S 


7. 1 249 7. 1 244 7. 1 238 7. 1 232 7. 1 226 


7.1221 7.12157.12097.12037.1198 


3 


3 


.6 


H92 n86 1 180 1 1 75 1 169 


1163 1158 1152 1146 1141 


4 


3 


•7 


1135 1129 1124 1118 1113 


1107 1101 1096 1090 1085 


4 


4 


.8 


1079 1073 1068 1062 1057 


1051 1046 1040 1035 1029 


s 


4 


•9 


1024 1018 1013 1007 1002 


0996 0991 0985 0980 0975 


5 


5 


8.0 


7.0969 7.0964 7.0958 7.0953 7.0947 


7.0942 7.0937 i^-°93i 7.0926 7.0921 


-6 


-5 


.1 


0915 0910 0904 0899 0894 


0888 0883 0878 0872 0867 


1 


I 


.2 


0862 0857 0851 0846 0841 


0835 0830 0825 0820 0814 


1 


I 


•3 


0809 0804 0799 0794 0788 


0783 0778 0773 0768 0762 


2 


2 


•4 


0757 0752 0747 0742 0737 


0731 0726 0721 0716 0711 


2 


2 


8.S 


7.0706 7.0701 7.0696 7.0691 7.0685 


7.0680 7.0675 7.0670 7.0665 7.0660 


3 


3 


.6 


0655 0650 0645 0640 0635 


0630 0625 0620 0615 0610 


4 


3 


•7 


0605 0600 0595 0590 0585 


0580 0575 0570 0565 0560 


4 


4 


.8 


0555 0550 0545 0540 0535 


0531 0526 0521 0516 0511 


5 


4 


■9 


0506 0501 0496 0491 0487 


0482 0477 0472 0467 0462 


S 


5 


9.0 


7.0458 7.0453 7.0448 1.0443 7.0438 


7.0434 7.0429 7.0424 7.0419 7.0414 


-5 


-4 


.1 


0410 0405 0400 0395 0391 


0386 0381 0376 0372 0367 


I 





.2 


0362 0357 0353 0348 0343 


0339 0334 0329 0325 0320 


1 


1 


•3 


0315 0311 0306 0301 0297 


0292 0287 0283 0278 0273 


2 


1 


•4 


0269 0264 0259 0255 0250 


0246 0241 0237 0232 0227 


2 


2 


9.5 


7.0223 7.0218 7.0214 7.0209 7.0205 


7.02007.0195 7.0191 7.01867.0182 


3 


2 


.6 


0177 0173 0168 0164 0159 


0155 0150 0146 0141 0137 


3 


2 


•7 


0132 0128 0123 0119 0114 


0110 0106 0101 0097 0092 


4 


3 


.8 


0088 0083 0079 0074 0070 


0066 0061 0057 0052 0048 


4 


3 


•9 


0044 0039 0035 0031 0026 


0022 0017 0013 0009 0004 


S 


4 



(7) 



4 PL. COLOGS. 



5 PL. LOGS. 
ABBREV, TAB. 



TABLE OF FIVE PLACE LOGARITHMS, 



CONTAINING 



An Abbreviated Table for One and Two Place Numbers ; 
A Table for Five Place Numbers from 1.0 to I.I Avoiding Interpolation ; 
A Table for All Four Place Numbers with Interpolation Tables for the 
Fifth Place. 



No. 


log. 


No. 


log. 


No. 


log. 


No. 


log. 


No. 


log. 





— 00 


2.0 


.30 103 


4.0 


.60206 


6.0 


.77815 


8.0 


.90309 


I 


.ooooo 


2.1 


.32 222 


4-1 


.61 278 


6.1 


•78 533 


8.1 


.go 849 


2 


.30 103 


2.2 


.34242 


4.2 


•62 325 


6.2 


•79 239 


8.2 


.91 381 


3 


•47 7»2 


2.3 


•36 173 


4-3 


•63 347 


6^3 


•79 934 


8.3 


.91 908 


4 


.60206 


2.4 


.38021 


4.4 


•64345 


6.4 


.80618 


8.4 


.92 428 


5 


.69 897 


2.5 


•39 794 


4.5 


.65 321 


6.5 


.81 291 


8.5 


.92 942 


6 


.77815 


2.6 


.41 497 


4.6 


.66 276 


6.6 


.81 954 


8.6 


. ^93450 


7 


.84510 


2.7 


.43 136 


47 


.67 210 


6.7 


.82607 


8.7 


■93 952 


8 


.90309 


2.8 


.44716 


4.8 


.68 124 


6.8 


.83251 


8.8 


.94448 


9 


•95 424 


2.9 


.46 240 


4'.9 


.69 O30 


6.9 


.83885 


8.9 


•94 939 


1.0 


.00000 


3.0 


.47712 


5.0 


.69 897 


7.0 


.84510 


9.0 


■95 424 


I.I 


.04 139 


3-1 


•49 136 


5^1 


•70 757. 


7^1 


.85 126 


91 


•95 904 


1.2 


.07 918 


32 


■SOS'S 


5-2 


.71 600 


7.2 


•85 733 


9.2 


•96 379 


1-3 


•"394 


3-3 


.51851 


5^3 


.72 428 


7.3 


.86332 


9^3 


.96 848 


1.4 


.14613 


3-4 


•53 148 


5'4 


•73 239 


7-4 


•86 923 


9.4 


•97313 


i-S 


.17609 


3^5 


•54407 


5^5 


•74036 


7-5 


.87 506 


9-5 


•97 772 


1.6 


.20412 


3-6 


•55 630 


5.6 


•74819 


7.6 


.88081 


9.6 


.98 227 


1-7 


.23045 


3-7 


.56 8?o 


5-7 


•75 587 


7-7 


.88 649 


9^7 


•98 677 


1.8 


.25 527 


3^8 


•57 978 


5.8 


•76 343 


7.8 


.89 209 


9.8 


•99 123 


1-9 


•27875 


3^9 


.59 106 


5-9 


•77 085 


7.9 


■89 763 


9-9 


■99 564 



(9) 



5 PL. LOGS. 
ABBREV. TAB. 



1.0-1.1. 
LOGS. 5 PL. 



FIVE PLACE LOGARITHMS. 



No. 


01234 


5 6 


7 


8 


9 


1.000 


.00000 .00004 .00009 .00013 .00017 


.00022 .00026 


.00030 


.00035 


.00039 


.OOI 


oo'043 00 048 00 052 00 056 00 o6i 


00 065 00 069 


00074 


00078 


00082 


.002 


00087 00091 00095 00 100 00104 


00108 00113 


00 117 


00 121 


00126 


.003 


00 130 00 134 00 139 00 143 00 147 


00 152 00 156 


00 160 


00 165 


00169 


.004 


00173 00178 00182 00 186 00 191 


00 195 00 199 


00204 


^00 208 


00212 


1.005 


.QP217 .00221 .00225 .00230 .00234 


.00238 .00243 


.00 247 


.00251 


-00255 


.006 


00 260 00 264 00 268 00 273 00 277 


00 281 00 286 


00290 


00294 


00 299 


.00/ 


00303 00307 00312 00316 00320 


00325 00329 


00333 


00337 


00342 


.008 


00346 00350 00355 00359 00363 


00 368 00 372 


00376 


00381 


00385 


.009 


00 389 00 393 00 398 00 402 00 406 


00411 00415 


00419 


00424 


00428 


1.010 


.00432 .00436 .00441 .00445 -00449 


.00454 .00458 


.00462 


.00467 


.00471 


.oil 


00475 00479 00484 00488 00492 


00497 00501 


00505 


00509 


00514 


.012 


0051& 00522 00527 00531 00535 


00 540 00 544 


00548 


00552 


°os57 


.013 


00 561 00 565 00 570 00 574 00 578 


00 582 00 587 


00591 


00595 


00600 


.014 


00604 00608 00612 00617 00621 


00 625 00 629 


00634 


00638 


00642 


I.OI5 


.00647 -00651 .00655 .00659 .00664 


.00668 .00672 


.00677 


.00681 


.00685 


.016 


00 689 00 694 00 698 00 702 00 706 


00711 00715 


00719 


00724 


00728 


.017 


00 732 00 736 00 741 00 745 00 749 


00753 00758 


00762 


00766 


00771 


.oi8 


. 00 775 00 779 00 783 00 788 00 792- 


00 796 00 800 


00805 


00809 


00813 


.019 


00817 00822 00826 00830 00834 


00839 00843 


00847 


00852 


00856 


1.020 


.00 860 .00 864 .00 869 .00 873 .00 877 


.00881 .00886 


.00890 


.00894 


.00898 


.021 


00903 00907 00911 00915 00920 


00924 00928 


00932 


00937 


00941 


.022 


00945 00949 00954 00958 00962 


00966 00971 


00975 


00979 


00983 


.023 


00 988 00 992 00 996 01 000 01 005 


01 009 01 013 


01 017 


01 022 


01 026 


.024 


01 030 01 034 01 038 01 043 01 047 


01051 01055 


01 060 


01 064 


01 068 


1.025 


.01 072 K)i 077 .01 081 .01 085 .01 089 


.01 094 .01 098 


.01 102 


.01 106 


.01 111 


.026 


01 115 01 119 01 123 01 127 01 132 


01 136 01 140 


01 144 


01 149 


01 153 


.027 


01 157 01 161 01 166 01 170 01 174 


01 178 01 182 


01 187 


01 191 


01195 


.028 


01 199 01 204 01 208 01 212 01 216 


01 220 01 225 


01 229 


01233 


01237 


.029 


01 242 01 246 01 250 01 254 01 258 


01 263 01 267 


01 271 


01275 


01 280 


1.030 


.01 284 .01 288 .01 292 .01 296 .01 301 


.01 305 .01 309 


.01 313 


.01 317 


.01 322 


.031 


01 326 01 330 01 334 01 338 01 343 


01347 01351 


01 355 


01360 


01364 


.032 


01 368 01 372 01 376 01 381 01 385 


01389 01393 


01397 


01 402 


01406 


•033 


01 410 01414 01418 01423 01427 


01 431 01435 


01439 


01444 


01 448 


■034 


01 452 01 456 01 460 01 465 01 469 


01473 01477 


oj 481 


01 486 


01 490 


I -035 


.01 494 .01 498 .01 502 .01 507 .01 511 


.01 515 .01 519 


-01 523 


.01 528 


.01 532 


.036 


01 536 01 540 01 544 01 549 01 553 


01 557 01 561 


01565 


01570 


01574 


■037 


01 578 01 582 01 586 01 590 01 595 


01 599 01 603 


01 607 


01 611 


01 616 


.038 


01 620 01 624 oi 628 01 632 01 636 


01 641 01 645 


01 649 


01653 


01657 


•039 


01 662 01 666 01 670 01 674 OI 678 


01 682 01 687 


01 691 


01 695 


01699 


1.040 


.01 703 .01 708 .01 712 .01 716 .01 720 


.01 724 .01 728 


•01 733 


•01 737 


.01 741 


.041 


01 745 01 749 01 753 01 758 01 762 


01 766 01 770 


01774 


01 778 


01783 


.042 


01 787 01 791 01 795 01 799 oi 803 


01 808 01 812 


01 816 


01 820 


01 824 


■043 


01 828 01 833 01 837 01 841 01 845 


01 849 01 853 


01858 


01862 


01866 


.044 


01 870 01 874 01 878 01 883 01 887 


01 891 01 895 


01899 


01903 


01907 


I -045 


.01 912 .01 916 .01 920 .01 924 .01 928 


.01 932 .01 937 


.01 941 


■01 945 


.01 949 


.046 


01 953 01 957 01 961 01 966 01 970 


01 974 01 978 


01 982 


01 986 


01 991 


.047 


01995 01999 02003 02007 02 01 1 


02015 02020 


02024 


02028 


02032 


.048 


02036 02040 02044 02049 02053 


02057 02061 


02065 


02069 


02073 


.049 


02 078 02 082 02 086 02 ogo 02 094 


02 098 02 102 


02 107 


02 III 


02 115 



LOGS. 5 PL. 
1.0-1.1. 



(10) 



FIVE PLACE LOGARITHMS. 



1.0-1.1. ' 

5 PL. LOGS. 



No. 



8 



1.050 

.051 
.052 

•053 
.054 

1-055 
.056 

■057 
.058 

•059 

1.060 

.061 
.062 
.063 
.064 

1.065 
.066 
.067 
.068 
.069 

1.070 

,.071 
.072 

•073 
.074 

1.075 
.076 
.077 
.078 
.079 

1.080 

.081 
.082 
.083 
.084 

1.08s 
.086 
.087 
.088 
.089 

1.090 

.091 
.092 

■093 
.094 

1.095 
.096 
.097 
.098 
.099 



.02119 .02123 .02127 .02131 .02135 

02 160 02 164 02 169 02 173 02 177 

02 202 02 205 02 210 02 214 02 2l8 

02243 02247 02251 02255 02259 

02 284 02 288 02 292 02 296 02 301 

.02325 .02329 .02333 .02338 .02342 

02366 02371 02375 02379 02383 

02407 02412 02416 02420 02424 

02449 02453 02457 02461 02465 

02 490 02 494 02 498 02 502 02 506 

.02531 .02535 .02539 .02543 .02547 

02 572 02 576 02 580 02 584 02 588 

02612 02617 02621 02625 02629 

02 653 02 657 02 661 02 666 02 670 

02 694 02 698 02 702 02 706 02 710 

.02735 .02739 .02743 .02747 .02751 

02 776 02 780 02 784 02 788 02 792 

02816 02821 02825 02829 02833 

02 857 02 861 02 865 02 869 02 873 

02898 02902 02906 02910 02914 

.02938 .02942 .02946 .02951 .02955 

02 979 02 983 02 987 02 991 02 995 
03019 03024 03028 03032 03036 

03 060 03 064 03 068 03 072 03 076 
03100 03104 03109 03113 03117 



•03 157 
03197 
03238 
03278 
03318 



.03 141 .03 145 .03 149 .03 153 

03 181 03 185 03 189 03 193 

03 222 03 226 03 230 03 234 

03 262 03 266 03 270 03 274 

03302 03306 03310 03314 

•03 342 
03383 
03423 
03463 
03503 

•03543 -03 547 •03551 -03555 -03559 

03583 03587 03591 03595 03 599 

03623 03627 03631 03635 03639 

03663 03667 03671 0367s 03679 

03703 03707 03711 03715 03719 

-03743 -03747 -03751 -03755 -03759 

03782 03786 03790 03794 03798 

03 822 -03 826 03 830 03 834 03 838 

03 862 03 866 03 870 03 874 03 878 

03902 03906 03910 03914 03918 

.03941 .03945 .03949 .03953 .03957 

03981 03985 03989 03993 03997 

04021 04025 04029 04033 04036 

04060 04064 04068 04072 04076 
04100 04104 04108 04112 04116 



.03 346 .03 350 .03 354 .03 358 
03387 03391 03395 03399 
03427 03431 0343s 03439 
03467 03471 0347s 03479 
03507 03511 03515 03519 

-03547 -03551 
03587 03591 
03627 03631 
03667 03671 
03707 03711 

-03747 -03751 

03786 03790 

•03 826 03 830 

03 866 03 870 

03 906 03 910 



.02 140 .02 144 .02 148 .02 152 .02 156 

02 181 02 185 02 189 02 193 02 197 

02 222 02 226 02 230 02 23s 02 239 

02 263 02 268 02 272 02 276 02 280 

02305 02309 02313 02317 02321 

.02 346 .02 350 .02 354 .02 358 .02 362 
02387 02391 02395 02399 02403 
02428 02432 02436 02440 02444 
02469 02473 02477 02481 02485 
02510 02514 02518 02522 02526 

.02551 .02555 .02559 .02563 .02567 
02 592 02 596 02 600 02 604 02 608 
02633 02637 02641 02645 02649 
02 674 02 678 02 682 02 686 02 690 
02715 02719 02723 02727 02731 

.02 755 .02 759 .02 763 .02 768 .02 772 
02796 02800 02804 02808 02812 
02 837 02 841 02 845 02 849 02 853 
02 877 02 882 02 886 02 890 02 894 
02918 02922 02926 02930 02934 



.02 959 
02999 
03040 
03080 
03 121 

.03 161 
03 201 
03242 



.02 963 
03003 
03044 
03084 
03125 

.03 165 
03205 
03246 



03 282 03 286 
03322 03326 



.02 967 
03007 
03048 
03088 
03129 

.03169 
03209 
03250 
03 290 
03330 



.02 971 
03011 
03052 
03092 
03133 

•03 173 
03214 

03254 
03294 
03334 



-02 975 
03015 
03056 
03096 
03137 

-03 177 
03 218 
03258 
03298 
03338 



.03 362 .03 366 .03 371 .03 375 .03 379 

03403 03407 03411 03415 03419 

03443 03447 03451 03455 03459 

03483 03487 03491 03495 03499 

03523 03527 03531 03535 03539 

•03563 -03567 •03571 •OS 575 -03579 

03603 03607 03611 03615 03619 

03643 03647 03651 03655 03659 

03683 03687 03691 03695 03699 

03723 03727 03731 03735 03739 

.03 763 -03 767 -03 771 •OS 775 -03 778 
03 802 03 806 03 810 03 814 03 818 
03842 03846 03850 03854 03858 

03 882 03 886 03 890 03 894 03 898 
03922 03926 93930 03933 03937 

.03961 .03965 .03969 .03973 .03977 
04001 04005 04009 04013 04017 
04040 04044 04048 04052 04056 
04080 04084 04088 04092 04096 

04 120 04 123 04 127 04 131 04 135 

5 PL. LOGS. 
1.0-1.1. 



1. 

LOGS. 5 PL. 



FIVE PLACE LOGARITHMS. 



No. 



8 



1.00 

.OI 
.02 

•03 

.04 

I. OS 

.06 
.07 
.08 
.09 

1.10 

.II 

.13 

•13 

.14 
i.iS 

.16 

•17 
.18 
.19 

1.20 

.21 
.22 

•23 
.24 

1. 25 
.26 
.27 
.28 
.29 

1.30 

■31 

■P 
•33 
•34 

'•35' 
•36 
■37 
•38 
■39 

1.40 

.41 
.42 
■43 
■44 

'•45 
.46 

■47 
.48 
•49 



.00 000 

00432 

00860 

01 284 
01703 

.02 119 

02 53-<, 
02938 

03342 

03743 

■04 139 
04532 
04922 
05308 
05690 

.06 070 
06446 
06819 
07188 
0755s 

.07918 
08279 
08636 
08991 
09342 

.09691 
10037 
10380 

10 721 
II 059 

•II 394 

11 727 

.i?.°S7 
12385 
12710 

•13033 
13354 
13672 
13988 
14 301 



.00043 
00475 
00903 
01 326 
01745 

.02 160 
02572 
02979 

03383 
03782 

.04179 
04571 
04961 
05346 
05729 
.06 108 
06483 
06856 
07225 
07591 

.07954. 
08314 
08672 
09026 
09377 
.09 726 . 
10072 

10415 
10.755 
1 1 093 

.1 1 428 , 
II 760 
12090 
12418 
12743 

.13066 . 
13386 
13704 
14019 

14333 



.00087 
00518 
00945 
01 3(18 

01 787 

.02 202 

02 612 

03 019 

03423 
03 822 

.04 21^ . 

04610 

04999 

05385 
05767 

.06 145 . 

06 521 
06893 

07 262 
07628 



,00 130 . 

00 561 
00988 

01 410. 
01 828 

.02 243 , 

02653 

03060 

03463 
03862 



.08 027 

08386 

08743 
09096 
09447 

•09 795 

10 140 
10483 
10823 

11 160 

•II 494 

11 826 

12 156 
12483 
12808 

• 13 130 
13450 

13767 
14082 

14395 



.14675. 

14983 

15 290 

15594 
15897 
.16 197 . 

16495 

16 791 

17085 
17377 



.00346 

00 775 

01 199 

01 620 

02 036 



.04 297 
04689 
05077 
05 46 r 
05843 
.06221 
06595 
06967 

07335 

07 700 

.08063/ 

08 422/ 
08778 

09 132 

09 482 

.09 830 

10 175 
10517 
10857 

11 193 

.11528 

11 860 

12 189 
12516 
12840 

.13162 
13481 

13799 
14 114 
14426 

'•14737 

: 15045 

' 15 351 

15655 

' 15957 

.16 256 

16554 

16 850 

17 143 

1743s 



•04 376 
04 766 

05154 
05538 
05918 

.06 296 
06670 
07041 
07408 
07773 



.11 561 
11893 

12 222 
12548 
12872 

•13 194 

13 513 
13830 

14 145 
14457 

.14768 , 
15076 

15 381 
15685 
15987 

.16286. 
16584 
16879 

17 173 
17464 



■II 594 
II 926 
12254 
12581 
12905 

.13 226 

13545 
13862 
14176 
14489 

14 799 

15 106 
15412 

15 715 

16 017 

16316 

16 613 
16909 

17 202 
17493 



04415 
04805 

05 192 
05576 
05956 
■06 '333 

06 707 
07078 

07445 
07809 

.08171 
08529 
08884 
I 09237 
09587 

■09 934 
10278 

10 619 
■10 958 

11 294 

.11628 

"959 
12287 

12 613 
12937 

.13258 
13577 
13893 
14208 
14520 

.14829 

15137 
15442 

15746 
16047 

.16 346 
16643 
16938 
17231 
17 522 



.04454 
04844 
05231 
05614 
05994 
■06371 
06744 

0711$ 
07482 
07 846 

.08 207 
08565 
08920 

09 272 
09621 

.09 968 

10 312 
10653 
10992 
11327 



,00 389 
00817 
01 242 

01 662 
02078 
.02 490 

02 898 
03302 

03703 

04 100 

.04493 
04883 

05 269 
05 652 
06032 



.08 243 
08600 
08955 
09307 
09656 
.10 003 
10346 
10687 
11025 
II 361 



11 661 .11 694 
11992 12024 
12320 12352 

12 646 12 678 
12969 13,001 

13290.13322 
13609 13640 

13925 13956 
14239 14270 
14551 14582 



14860 
15168 
15473 
15776 
16077 

16376 
16673 
16967 
17 260 
17 551 



.14891 

15 198 

15503 
15806 

16 X07 

.16406 
16 702 
16997 
17289 
17580 



INTERPO. 


lABLES. 


ForlogLO 


tologl.l 


interpolated 


values are 


Kiven on thf 


preceding 


two pages. 


38 36 


4 4 


8 7 


II II 


15 14 


19 18 


23 22 


27 25 


30 29 


34 32 


34 33 


3 3 


7 7 


10 ID 


14 13 


17 17 


20 20 


24 23 


27 26 


31 30 


32 31 


.3 3 


6 6 


10 9 


13 12 


16 16 


19 19 


22 22 


26 25 


29 28 


30 29 


3 3 


6 6 


9 9 


12 12 


15 15 


18^17 


21 20 


24 23 


27 26 



LOGS. 5 PL. 



(12) 



FIVE PLACE LOGARITHMS. 





^_,__ 


6 PL. LOGS. 




No. 


01234 56789 


INTERPO 
TABLES. 






1.50 


.17 609 .17 638 .17 667 .17 696 .17 725 .17 754 .17 782 .17 811 .17 840 .17 869 


29 27 


.00 




•51 


17898 17926 17955 17984 18013 18041 18070 18099 18 127 18156 


3 3 


.30 




•52 


18 184 18213 18 241 18270 18298 18327 18355 '8384 18 412 18 441 


6 5 






■53 


18469 18498 18526 18554 18583 18 611 18639 18667 18696 18724 


9 8 






•54 


18752 18780 18808 18837 18-865 18893 18 921 18949 18977 19 008 


12 11 






"•55 


.19033 .19061 .19089 .19117 .19145 .19173 .19201 .19229 .19257 .19285 


15 14 






•55 


19312 19340 19368 19396 19424 19451 19479 19507 19535 19562 


17 16 






•57 


19590 19618^9645 19673 19700 19728 19756 19783 19 811 19838 


20 19 






•58 


19866 19 893° 19 921 19948 19976 20 003 20030 20058 20085 20112 


23 22 






•59 


20140 20167 20194 20222 20249 20276 20303 20330 20358 20385 


26' 24 






1.60 


.20 412 .20 439 .20 466 .20 493 .20 520 .20 548 .20 575 .20 602 .20 629 .20 656 


26 25 






.61 


20683 20710 20737 20763 20790 20817 20844 20371 20898 20925 


3 3 






.62 


20952 20978 21005 21032 21059 21085 21 112 21139 211^5 21192 


5 5 






•63 


21219 21245 21272 21299 21325 21352 21378 21405 21431 21458 


8 8. 






.64' 


21484 21 511 21537 21564 21590 21617 21643 21669 21696 21722 


10 ih 






1.6s 


, .21 748 .21 775 .21 801 .21 827 .21 854 .21 880 .21 906 .21 932 .21 958 .21 985 


13 13 






.66 


22 011 22037 22063 22089 22115 22 141 22167 221^4 22220 22246 


16 15 






.67 


22272 22298 22324- 22350 22376 22401 22427 22453 22479 22505 


18 18 






.68. 


22531 22557 22583 22608 22634 22660 22686 22712 22737 22763 


21 20 






.69 


22789 22814 22840 22866 22891 22917 22943 22968 22994 23 019 


23 23 






1.70 


.23 045 .23 070 .23 096 .23 121 .23 147 .23 172 .23 198 .23 223 .23 249 .23 274 


25 24 






•71 


23300 23325 23350 23376 23401 23426 23452 23477 23502 23528 


3 2 






.72 


23553 23578 23603 23629 23654 23679 23704 23729 23754 23779 


5 5 






■73 


23S05 23830 23855 23880 23905 23930 23955 23980 24 005 24030 


8 7 






■74 


24055 24080 24105 24130 24155 24180 24204 24229 24254 24279 


10 10 






'•75 


.24 304 .24 329 .24 353 .24 378 .24 403 .24 428 .24 452 .24 477 .24 502 .24 527 


13 12 






.76 


,24551 24576 24601 24625 24650 24674 24699 24724 24748 24773 


15 14 






•77 


24797 24822 24846 24871 24895 24920 24944 24969 24993 25 018 


18 17 






•78 


25 042 25 066 25 091 25 115 25 139 25 164 25 188 25 212 25 237 25 261 


20 19 






•79 


25285 25310 25334 25358 25382 25406 25431 25455 25479 25503 


23 22 






1.80 


■25 527 ^25 551 ^25 575 ^25 600 .25 624 .25 648 .25 672 .25 696 .25 720 .25 744 


24 23 






.81 


25 768 25 792 25 816 25 840 25 864 25 888 25,9,12 25 935 25 959 25 983 


2 2 






.82 


26 007- 26 031 26055 26079 26102 26126 26150 26174 26198 26221 


5 5 






•83 


26245 26269 26293 26316 26340 '26364 26387 26411 26435 26458 


7 7 






.84 


26482 26505 26529 26553 26576 26600 26623 26647 26670 26694 


10 9 






1.8s 


.26717 .26741 .26764 .26788 .26811 .26834 .26858 .26881 .26905 .26928 


12 12 






.86 


26951 26975 26998 27 021 27045 27068 27091 27114 27138 27161 


14 14 






•87 


27 184 27 207 27 231 27 254 27 277 27 300 27 323 27 346 27 370 27 393 


17 16 






.88 


27416 27439 27462 27485 27508 27531 27554 27577 27600- 27 623 


19 18 






.•89 


27646 27669 27692 27715 27738 27761 27784 27807 27830 27852 


22 20 






1.90 


.27 875 .27 898 .27 921 .27 944 .27 967 .27 989 .28 012 .28 035 .28 058 .28 081 


22 21 






•91 


28103 28126 28149 28 171 28194 28217 28240' 28262 28285 28307 


2 2 






.92 


28330 28353 28375 28398 28421 28443 28466 28488 28511 28533 


4 4 






•93 


28556 28578 28601 28623 28646 28668 28691 28713 28735 28758 


7 6 






■94 


28780 28803 28825 28847 28870 28892 28914 28937 28959 28981 


9 8 






■95 


.29 003 .29 026 .29 048 .29 070 .29 092 .29 115 .29 137 .29 159 .29 181 .29 203 


11 11 






.96 


29226 29248 292.70 29292 29314 29336 29358 29380 29403 29425 


13 13 


00 




■97 ■ 


29447 29469 29491 29513 29535 29557 29579 29601 29623 29645 


15 15 


.30 




.98 


29667 29688 29710 29732 29754 29776 29798 29820 29842 29863 


18 17 






■99 


29885 29907 29929 29951 29973 29994 30016 30038 30060 30081 


20 19 








(13) 5 PL 


.. LOG 
1. 


s. 



2. 
LOGS. 6 PL. 



FIVE PLACE LOGARITHMS. 



.30 

.47 



.80 

.47 



No. 



2.00 

.OI 
.02 

•03 

.04 

2.05 

.06 
.07 
.08 
.09 

2.10 

.II 

.12 

•13 
.14 

2.15 
.16 

•17 
.18 



.19 

2.20 

.21 
.22 

•23 
.24 

2.25 
.26 
.27 
.28 
.29 

2.30 

•31 
•32 
•33 
•34 

2-35 
•36 
•37 
.38 
•39 

2.40 

.41 
.42 
•43 

■44 

2-45 
.46 

■47 
.48 
•49 



8 



LOGS. 
2. 



•30 103 
30320 

30535 
30750 
30963 

•31 175 
31387 
31597 
31806 

32 015 

.32 222 
32428 
32634 
32838 
33041 

•33244 

33 445 
33646 
33846 
34044 

•34 242 
34439 
34635 
34830 

35025 
•35 218 
35 411 
35603 

35 793 
35984 

•36 173 
36361 
36549 
36736 
36922 

•37 107 
37291 

37 475 
37658 
37840 

.38 021 

38 202 
38382 
38561 
38739 

•38917 
39094 
39270 

39 445 
39 620 

6 PL. 



■30 125 
30341 
30557 
30771 
30984 

•31 197 
31408 
31 618 
31827 
32035 

■32 243 
32449 
32654 
32858 
33062 

•33 264 
33465 
33666 
33866 
34064 

•34 262 
34459 
3465s 
34850 
35044 
•35 238 
35430 
35622 

35813 
36 003 



.30 146 , 

30363 

30578 

30792 

31006 

.31 218 
31429 
31639 
31848 
32056 

.32 263 

32469 
3267s 
32879 
33082 

•33 284 
33486 
33686 
33885 
34084 

,34 282 

34 479 
34674 
34869 

35064 

35 257 
35 449 
35641 
35832 
36021 



.36 192 .36 211 
36380 36399 



36568 
36754 
36940 

•37 J2S 
37310 

37 493 
37676 

37858 

■38039 

38 220 

38399 
38578 
38757 

■38 934 

39 III 
39287 
39463 
39637 



36586 
36773 
36959 

■37144 
37328 
37 5" 
37694 
37876 

•38057 
38238 
38417 
38596 
38775 

•38 952 
3? 129 

39305 
39480 

39655 



,30 168 

30384 

30 600 
30814 
31027 

•31 239 
31450 

31 660 
31869 
32077 

.32 284 
32490 
32695 
32899 
33102 

•33 304 
33506 
33706 
33905 
34104 

•34 301 
34498 
34694 
34889 
35083 

•35 276 
35468 
35660 

35851 
36040 

.36 229 
36418 
3660s 
36791 
36977 
•37 162 
37346 
37530 
37712 
37894 

■3807s 
38256 

3843s 
38614 
38792 

■38 970 
39146 
39322 
39498 
39672 



■30 190 
30406 
30621 
30835 
31048 

.31 260 

31471 
31681 
31890 
32098 

•32305 
32510 

32715 
32919 
33122 

■33 325 
33526 
33726 
33925 
34124 

•34 321 
34518 
34713 
34908 
35102 

•35 295 
35488 
35679 
35870 
36059 

.36 248 

36436 
36624 
36810 
36996 

•37 181 
37365 
37548 
37 731 
37912 

•38093 
38274 
38453 
38632 
38810 

.38 987 
39164 
39340 

39515 
39690 

(14) 



.30211 , 

30428 

30643 

30856 

31069 

.31 281 
31492 
31 702 

31 911 

32 118 

•32 325 
32531 
32736 
32940 
33143 

•33 345 
33546 
33746 

33 94S 
34143 

•34 341 

34 537 

34 733 
34928, 

35 122 

•35 315 
35507 
35698 
35889 
36078 

•36 267 

36455 
36642 
36829 
37 014 

■37 199 
37383 
37566 
37 749 
37931 

.38112 
38292 

38471 
38650 
38828 

.39 005 

39182 
39358 
39 533 
39707 



•30 255 

30471 
30685 

30899 

31 112 

■31 323 
31534 
31744 
31952 

32 160 

.32 366 
32572 
32777 
32980 
33183 

•33 385 
33586 
33786 
33985 
34183 



•30 233 
30449 
30664 
30878 
31091 
.31 302 
31 513 
31723 
31931 
32139 

■32 346 
32552 
32756 
32960 

33163 

•33 365 
33566 
33766 
33965 
34163 

•34 361 
34 557 
34 753 

34 947 

35 141 

•35 334 ^35 353 
35526 35 545 
35717 35736 
35908 35927 
36097 36 116 

.36286 .36305 

36474 36493 
36661 36680 
36847 36866 
37033 37051 
.37 218 .37 236 
37401 37420 
37585 ^7603 
37 767 1 37 785 
37 949 37967 

.38 130 .38 148 
38310 38328 

38489 38507 
38668 38686 
38846 38863 

.39023^.39041 
39 199 39217 
39 375 39 393 
39550 39568 
39724 39742 



.30 276 
30492 
30707 
30920 

31 133 

•31 345 
31555 
31765 
31973 

32 181 

•32 387 
32593 
32797 

33 001 
33203 

•33405 
33606 
33806 

34 005 
34203 



•34380 ^34400 

34 577 34596 
34772 34792 
34967 34986 

35 160/ 35 180 

•35 372 
35564 
35 755 
35946 
36135 



■30 298 
30514 
30728 
30942 
31 154 
■31 366 
31576 
31785 
31994 
32201 

•32 408 
32613 
32818 
33021 
33224 

•33425 
33626 
33826 
34025 
34223 

.34420 
34616 

34 81 1 

35 005 
3S»i99 

•35 392 
35583 
35 774 
35965 
36154 



.36324 ^36342 
36511 36530 
36698 36717 
36884 36903 
37070 37088 



•37 254 
37438 
37621 
37803 
37985 

.38 166 
38346 
38525 
38703 
38881 

•39 058 
39235 
39410 

3958s 
39 759 



•37 273 
37 457 
37639 
37822 
38003 

.38 184 
38364 
38543 
38721 

38899 
.39076 
39252 
39428 
39602 

39 777 



FIVE PLACE LOGARITHMS. 



5 PL. LOGS. 



No. 



8 



INTP, 
TAB. 



2.50 

•5' 

•52 
•S3 
•54 

^•55 
•56 
■57 
.58 

•59 

2.60 

.61 
.62 

•63 
.64 

2.65 
.66 
.67 
.68 
•69 

2.70 

•71 
.72 

■73 
•74 

2^75 
.76 

■77 
.78 

■79 

2.80 

.81 
.82 

■83 
.84 

2.85 
.86 

■87 
•.88 

■89 

2.90 

.91 
■92 
■93 
■94 

2.95 
.96 

•97 
.98 

■99 



•39 794 
39967 

40 140 
40312 
40483 

.40 654 
40824 

40993 

41 162 

41330 

.41 497 

41 664 
41830 
41996 

42 160 

•42 325 
42488 
42651 
42813 
42975 

•43 136 
43297 

43 457 
43616 

43 775 

•43 933 
44091 
44248 
44404 
44560 

•44 716 
44871 

45025 
45179 
45332 

•45 484 
45637 
45788 
45 939 
46090 

.46 240 
46389 
46538 
46687 
46835 

.46 982 
47129 
47276 
47422 

47567 



.39811 
39985 
40157 
40329 

40SSP 
.40671 
40841 
41010 
41 179 
41347 

.41 514 
41 681 
41847 
42012 

42177 

.42 341 
42504 
42-667 
42830 
42991 

•43 152 
43313 
43 473 
43632 
43791 

•43 949 
44107 
44264 
44420 
44576 

■44 731 
44886 

45040 
45194 
45 347 
•45500 
45652 
45803 

45 954 

46 105 

■46 25 s 
46404 

46553 
46702 
46850 

•46 997 
47144 
47290 
47436 
47582 



•39 829 

40 002 

40175 
40346 
40518 

.40 688 
40858 

41 027 
41 196 
41363 

•41 531 
41697 
41863 
42029 
42193 

•42 357 
42521 
42684 
42846 

43 008 

■43 169 
43329 
43489 
43648 
43807 

•43 965 

44 122 
44279 
44436 
44592 

•44 747 
44902 

45056 
45209 
45362 

■45 515 
45667 
45818 

45969 
46 120 

.46 270 
46419 
46568 
46716 
46864 

.47 012 

47159 
-47305 
47451 
47596 



■39 846 
40019 

40 192 
40364 
4P535 
.40 705 
40875 
41044 

41 212 
41 380 



■39 863 
40037 
40209 
40381 
40552 

.40 722 
40892 
41 061 
41 229 
41397 



.41 547 .41 564 
41 714 41 731 

41 880 41 896 

42 045 42 062 
42 210 42 226 



■42 374 
42537 

42 700 
42862 
43024 

•43 185 

43 345 
43505 
43 664, 
43823 

■43 981 
44138 
44295 

44451 
44607 



.42 390 

42553 

42 716 
42878 
43040 

■43 201 
43361 
43521 

43 680 
43838 

■43 996 
44154 

44 31 1 
44467 
44623 



.44762 .44778 
44917 44932 
45071 45086 
45 225 45 240 
45378 45 393 



■45-530 
45 682 
45834 
45984 
46135 

.46 285 
46434 
46583 
4673" 
46879 

.47 026 
47173 
47319 
47465 
47 61 1 



•45 545 
45697 
45849 

46 000 
46150 

.46300 
46449 
46598 
46746 
46894 

•47 041 
47188 

47 334 
47480 
47625 



.39 881 
40054 
40226 
40398 
40569 

•40 739 
40909 
41 078 
41 246 
41414 

••41 581 
41747 

41 913 

42 078 
42243 

.42 406 
42570 
42732 
42894 
43056 

•43217 

43 377 
43 537 
43696 

43854 
.44 012 

44170 
44326 

44483 
44638 

•44 793 
44948 
45 102 
45255 
45408 

■45 561 
45712 
45864 
46015 
46165 

•46315 
46464 
46613 
46761 
46909 

■47 056 
47 202 

47 349 
47 494 
47640 



■39 898 
40071 
40243 
40415 
40586 

.40 756 
40926 
41095 
41263 
41430 

•41 597 
41764 
41929 
42095 
42259 

•42 423 
42586 

42749 

42 911 
43072 

•43 233 

43 393 
43 553 
43712 
43870 
.44028 
44185 
44342 
44498 
44654 

.44 809 
44963 
45117 
45271 
45423 

•45 576 
45728 

45879 
46030 
46 180 

•46 330 
46479 
46627 

46776 
46923 

•47 070 
47217 

47363 
47509 
47654 



•39915 
40088 
40261 
40432 
40603 

■40 773 
40943 
41 III 
41 280 
41447 

.41 614 

41 780 
41946 

42 III 
42275 

•42 439 
42602 

42765 
42927 
43088 

•43 249 
43409 
43569 
43727 
43886 

•44044 
44 201 

44358 
44514 
44669 

.44 824 

44 979 
45133 

45 286 
45 439 

■45 591 
45 743 
45894 
46045 
46195 

•46 345 
46494 
46642 
46790 
46938 

.47085 
47232 
47378 
47524 
47669 



■39 933 

40 106 
40278 
40449 
40620 

•40 790 
40960 

41 128 

41 296 
41464 

■41 631 
41797 
41963 

42 127 

42 292 

■42 455 
42619 
42781 
42943 
43104 

■43 265 
43425 
43584 

43 743 
43902 

•44059 .44075 
44217 44232 

44 373 44389 
44529 44 545 
44 685 44 700 



•39 950 
40123 
40295 
40466 
40637 
.40 807 
40976 
41 145 
41313 
41 481 

.41 647 
41 814 
41979 
42144 
42308 

■42 472 
42635 
42797 

42959 
43120 

.43 281 
43441 
43600 

43 759 
43917 



.44840 

44 994 
45148 
45301 

45 454 
.45 606 
45758 

45 909 
46060 
46210 



.44855 
45 010 

45 163 
45317 
45469 

.45 621 
45 773 
45924 
46075 
46225 



(15) 



•46 359 -46 374 
46509 46523 
46657 46672 

46 805 46 820 
46953 46967 
.47 100 .47 114 

47 246 47 261 
47392 47407 
47538 47 553 
47683 47698 

6 PL. 



LOGS. 
2. 



FIVE PLACE LOGARITHMS. 



LOGS. 5 PL. 



No. 



8 



3.00 

.OI 

.02 

•03 
.04 

305 
.06 
.07 
.oS 
.09 

3.10 

.11 
.12 

•>3 
.14 

3-«5 
.16 

•17 
.18 

•19 

3.20 

.21 
.22 

•23 
.24 

3-25 
.26 
.27 
.28 
.29 

3.30 

■31 
•32 
■33 
•34 

3-35 
■36 
•37 
•38 
•39 

3.40 

•4' 
.42 

•43 
•44 

3-45 
.46 

•47 
.48 

•49 



LOGS. 
3. 



•47712 • 
47857 

48 001 

48144 
48287 

•48 430 
48572 
48714 
48855 
48996 

■49 136 
49276 

49415 

49 554 
49693 

•49 831 
49969 

50 106 
5° 243 
50379 

.50515 
50651 
50786 
50920 
51055 
.51 188 
51322 
51455 
51587 
51720 

.51 851 

5«983 
52 114 
52244 
52375 

■52 504 
52634 
52763 
52892 
53020 

■53 148 
53275 
53403 
53529 
53656 

■53 782 
53908 
54033 
54158 
54283 

6 PL. 



47 727 
47871 
48015 

48159 
48302 

.48444 
48586 
48728 
48869 

49 010 

•49 150 
49290 
49429 
49568 
49707 

•49 845 
49982 

50 120 
50256 
50393 

•50 529 
50664 

50799 
50934 
51068 

.51 202 

51335 
51468 

51 601 
51733 

.51865 
51996 
52127 
52257 
52388 

•52517 
52647 
52776 
52905 
53033 

■53 161 
53288 

53415 
53542 
53668 

■53 794 
53920 

54045 
54170 

54295 



■47 741 
47885 
48029 

48173 
48316 

.48458 
48601 
48742 
48883 
49024 

•49 164 
49304 

49 443 
49582 
49721 

■49 859 
49996 

50133 

50 270 
50406 

.50 542 
50678 
50813 

50947 

51 081 

.51215 
51348 
51481 
51 614 
5^746 

.51 878 
62 009 

52140 
52270 
52401 

•52 530 
52660 
52789 
52917 
53046 

■53 173 
53301 
53428 

53 555 
53681 

■53807 
53 933 
54058 
54183 
54307 



•47 756 

47900 

48044 

48187 

48330 

.48473 , 

48615 

48756 

48897 

49038 

•49 178 
49318 
49 457 
49596 

49 734 

•49 872 

50 010 

50147 
50284 
50420 

•50556 
50691 
50826 
50961 
51095 

.51 228 
51362 

51495 
51627 

51759 

.51 891 
52022 

52153 
52284 

52414 

•52 543 
52673 
52802 
52930 
53058 

•53 186 
53314 
53441 
53567 
53694 
•53 820 
53 945 
54070 

54195 
54320 



•47 770 
47914 
48058 
48202 
48344 

.48487 
48629 

48770 

48 9H 
49052 

•49 192 
49332 

49 471 
49610 
49748 

.49 886 
50024 

50 161 
50297 
50433 

•50 569 
50705 
50840 

50974 

51 108 

.51 242 

51375 
51508 

51 640 
51772 

.51 904 

52035 

52 166 

52297 
52427 
.52556 
52686 
52815 
52943 
53071 

•53 199 
53326 

53 453 
53580 
53706 
•53 832 
53958 
54083 
54208 
54332 



•47 784 
47929 
48073 
48216 
48359 
.48 501 
48643 
48785 
48926 
49066 



•47 799 
47 943 
48087 
48230 
48373 
.48515 
48657 
48799 
48940 
49080 



•47 813 
47958 
48 lOI 
48 244 
48387 
.48 530 
48671 
48813 
48954 
49094 



,47 828 .47 842 
47972 47986 
48 116 48 130 
48259 48273 

48 401 48 416 

.48 544 .48 558 
48686 48700 
48827 48841 
48968 48982 

49 108 49 122 



.49 206 .49 220 
49346 49360 
49485 49499 
49624 49638 
49762 49776 



.49900 
50037 
50174 

50 3" 
50447 

•50583 
50718 
50853 

50987 

51 121 

•51 255 
51388 
51 521 
51 654 
51786 

.51917 
52048 
52179 
52310 
52440 

•52 569 
52699 
52827 
52956 
53084 

.53212 

53 339 
53466 

53 593 
53719 

■53 845 
53970 
54095 
54220 

54 345 



•49 914 
50051 
50188 

50325 
50461 

.50 596 
50732 
50866 
51001 
51135 
.51 268 
51402 

51534 
51667 

51799 

•51 930 
52061 
52192 
52323 
52453 

•52 582 

52711 
52840 

52969 

53097 

•53 224 
53352 

53 479 
53605 

53732 

•53 857 
53983 

54 108 

54233 
54 357 



.49 234 .49 248 .49 262 
49 374 49388 49402 
49513 49527 49541 
49651 49665 49679 
49790 49803 49817 

•49927 ^49 941 ^49 955 
50065 50079 50092 
50202 50215 50229 
50338 50352 50365 
50474 50488 50501 



.50610 

50745 
50880 

51014 
51148 

.51 282 

51 415 
51548 
51680 
51812 

•51 943 
5-2075 
52205 

52336 
52466 

•52 595 
52724 
52853 
52982 
53 no 

•53 237 
53364 
53491 
53618 

53 744 

•53 870 

53 995 

54 120 
54245 
54370 



.50 623 
50759 
50893 

51 028 
51 162 

.51 295 
51428 
51561 
51693 
51825 

•51 957 
52088 
52218 
52349 
52479 
.52608 

52737 

52 866 
52994 
.53 122 

•53 250 

53 377 
53504 
53631 
53 757 
•S3 882 
54008 

54133 
54258 
54382 



•50 637 
50772 
50907 

51041 
pi75- 
•51 308 
51441 
51574 

51 706 
51838 

•51970 

52 lOI 

52231 
52362 

52492 
.52 621 
52750 

52879 

53 007 

53135 

•S3 263 
53390 
53517 
53643 
53769 

•53895 
54020 

54145 
54270 

54 394 



(16) 



FIVE PLACE LOGARITHMS. 







5 PL. 


LOGS 


No. 


01234 


5 6, 7 8 9 


INTP 
TAB. 




3.50 


.54407 ^54419 ^54432 ^54 444 .54 456 


•54469 ^54481 .54494 .54506 .54518 


13 


.4' 


•S' 


54 53" 54 543 54 555 54 568 54580 


54 593 54605 54617 54630 54642 


1 


.6C 


•52 


54654 54667 54679 54691 54704 


54716 54728 54741 54 753 54765 


3 




•S3 


54 777 54790 54802 54814 54827 


54839 54851 54864 54876 54888 


4 




•54 


54900 54913 54925 54 937 54 949 


54962 54 974 54986 54998 86 011 


5 




3-SS 


•55 023 .55 035 -55 047 •SS 060 .55 072 


.55084 .55096 .55 108 .55 121 .55 133 


7 




•56 


55 "45 "^55 157 55169 55182 55194 


55206 55218 55230 55242 55255 


8 




■57 


55267 55279 55291 55303 55315 


55328 55340 55352 55364 55376 


9 




■58 


55388 55400 55413 55425 55437 


55 449 55461 55473 55485 55497 


10 




•59 


55509 55522 55534 55546 55558 


55570 55582 55594 55606 55618 


12 




3.60 


•55 630 .55 642 .55 654 ^55 666 .55 678 


•55 691 -55 703 .55 715 •SS 727 ^55 739 


13 




.6i 


55751 55763 55775 55787 55799 


55811 ,55823 55835 55847 55859 


I 




.62 


55871 55883 55895 55907 55919 


55 931 55 943 55 955 55 967 55 979 


2 




■63 


55991 56 003 56015 56027 56038 


56050 56062 56074 56086 56098 


4 




.64 


56 no 56122 56134 56146 56158 


56170 56182 56194 56205 56217 


5 




3-65 


.56 229 .56 241 .56 253 .56 265 .56 277 


.56289 .56301 .56312 .56324 .56336 


6 




.66 


56348 56360 56372 56384 56396 


56407 56419 56431 56443 56455 


7 




.67 


56467 56478 56490 56502 56514 


56526 56538 56549 56561 56573 


8 




.68 


56585 56597 56608 56620 56632 


56 644 56 656 56 667 56 679 56 691 


10 




.69 


56703 56714 56726 56738 56750 


56761 56773 56785 56797 56808 


11 




3.70 


.56820 .56832 .56844 .56855 .56867 


.56879 .56891 .56902 .56914 .56926 


12 




•71 


56937 56949 56961 56972 56984 


56996 57 008 57019 57031 57043 


I 




.72 


57054 57066 57078 57089 57101 


57113 57124 57136 57148 57159 


2 




■73 


57171 57183 57194 57206 57217 


57229 57241 57252 57264 57276 


4 




•74 


57287 57299 57310 57322 57334 


57 345 57 357 57 368 57 380 57 392 


5 




3-75 


•57403 ^57415 ^57426 .57438 .57449 


■57461 .57473 .57484 .57496 .57507 


6 




.76 


57S"9 57530 57542 57553 57565 


57576 57588 57.600 57 6u 57623 


7 




•77 


57634 57646 57657 57669 57680 


57692 57703 57715 57726 57738 


8 




.78 


57 749 57761 57772 57784 57795 


57807 57818 57830 57841 57852 


10 




•79 


57864 57875 57887 57898 57910 


57 921 57933 57944 57955 57967 


11 




3.80 


■57978 ^57990 .58 001 .58013 .58024 


.58035 .58047 .58058 .58070 .58081 


11 




.81 


58092 58104 58115 58127 58138 


58 149 58 161 58 172 58 184 58 195 


1 




.82 


58 206 58 218 58 229 58 240 58 252 


58263 58274 58286 58297 58309 


2 




•83 


58320 58.331 58343 58354 58365 


58377 58388 58399 58410 58422 


3 




.84 


58433 58444 58456 58467 58478 


58490 58501 58512 58524 58535 


4 




3-85 


.58 546 .58 557 .58 569 .58 580 .58 591 


.58602 .58 614,. 58 625 .58636 .58647 


6 




.86 


58659 58670 58681 58692 58704 


58715 58726 58737 58749 58760 


7 




.87 


58 771 58 782 58 794 "58 805 58 816 


58827 58838 58850 58861 58872 


8 




.88 


58883 58894 58906 58917 58928 


58939 58950 58961 58973 58984 


9 




.89 


58995 59 006 59017 59028 59040 


59051 59062 59073 59084 59095 


10 




3.90 


.59 106 .59 118 .59 129 .59 140 .59 151 


.59 162 .59 173 .59 184 .59 195 .59 207 


11 




.91 


59218 59229 59240 59251 59262 


59273 59284 59295 59306 59318 


1 




.92 


59329 59340 59351 59362 59 373 


59384.59395 59406 59417 59428 


2 




•93 


59 439 59450 5946/ 59472 59483 


59 494 59506 59517 59528 59 539 


3 




.94 


59550 59561 59 572 59583 59 594 


59605 59616 59627 59638 59649 


4 




3-95 


.59 660 .59 671 .59 682 .59 693 .59 704 


■59 715 -59 726 .59 737 ^59 748 .59 759 


6 




.96 


59770 59780 59791 59802 59813 


59824 59835 59846 59857 59868 


7 


.47 


■97 


59879 59890 59901 59912 59923 


59 934 59 945 59 956 59 966 59977 


8 


.60 


.98 
■99 


59988 59999 60 010 60021 60032 


60043 60054 60065 60076 60086 


9 




60097 60108 60119 60130 60141 


60 152 60 163 60 173 60 184 60 195 


10 






(17) 6 PL. I 


.OG 
3. 


s. 



4. 
LOGS. 6 PL. 



FIVE PLACE LOGARITHMS. 



No. 



8 



4.00 

.01 
.02 

•03 
.04 

4-05 
.06 
.07 
.08 
.09 

4.10 

.11 
.12 

•13 

.14 

4.15 
.16 

•17 
.18 
.19 

4.20 

.21 
.22 

•23 
.24 

4.25 
.26 
.27 
.28 
.29 

4.30 

•3J 
■32 
•33 
•34 

4-35 
•36 
•37 
•38 
•39 

4.40 

.41 
.42 
■43 
•44 

4-45 
.46 

•47 
.48 
■49 



.60 206 
60314 
60423 
60531 
60638 

.60 746 
60853 
60959 
61 066 
61 172 

.61 278 
61384 
61 490 

61595 
61 700 

.61 805 

61 909 
62014 

62 118 

62 221 

.62 325 
62428 
62531 
62634 
62737 

•62 839 
62941 
63043 
63144 
63246 

•63 347 
63448 

63548 
63649 

63 749 

•63 849 
63949 
64048 
64147 
64246 

•64 345 
64444 
64542 
64640 
64738 

.64836 

64933 
65031 
65 128 
65225 



,60217 
60325 

60433 

60 541 
60649 

.60 756 
60863 
60970 

61 077 
61 183 

.61 289 

61395 
61 500 
61 606 
6i 711 

.61 815 
61 920 
62024 
62128 
62232 

•62.335 
62439 
62542 
62644 
62747 

.62 849 
62951 
63053 
63155 
63256 

•63 357 
63458 
63558 
63659 
63759 
.63 859 
63959 
64058 
64157 
64256 

•64355 
64454 
64552 
64650 
64748 

.64 84^ 

64943 
65040 

65137 
65234 



.60 228 , 

60336 

60444 

60552 

60660 

.60 767 
60874 
60981 
61087 
61 194 

.61 300 
61 405 
61 511 
61 616 

61 721 

.61 826 
61930 
62034 

62 138 
62242 

.62 346 
62449 
62552 
62655 
62757 

.62 859 
62961 
63063 
63165 
63266 

•63 367 
63468 
63568 
63669 
63769 

■63 869 
63969 
64068 
64167 
64266 

.64 365 
64464 
64562 
64660 
64758 
.64856 

64953 
65 050 
65147 
65244 



.60 239 
60347 
60455 
60563 
60670 

.60 778 
60885 
60991 
61 098 
61 204 

.61 3l[o 
61 416 
61 521 
61 627 
61 731 

.61 836 

61 941 
62045 

62 149 
62252 

•62 356 
62459 
62562 
62665 
62 767 

.62 870 
62972 
63073 
63175 
63276 

■63 377 
63478 
63579 
63679 

63779 

•63 879 
63979 
64078 
64177 
64276 

■64 375 
64473 
64572 
64670 
64768 

.64 865 
64963 
65 060 

65157 
65254 



.60 249 
60358 
60466 
60574 
60681 

.60788 
60895 
61002 
61 109 
61 215 

.61 321 
61 426 
61532 
61637 
61 742 

.61 847 
61 951 
62055 
62159 
62263 

.62 366 
62469 
62572 
62675 
62778 

.62 880 
62982 
63083 
63185 
63286 

■63387 
63488 
63589 
63689 

63789 
.63 889 
63988 
64088 
64187 
64286 

■64 385 
64483 
64582 
64680 
64777 

.64875 
64972 
65070 
65 167 
65263 



.60 260 
60 369 
60477 

60 584 
60692 

.60 799 
60906 

61 013 
61 119 
61 225 

■61 331 
61437 
61 542 
61648 
61 752 

.61 857 

61 962 
62066 

62 170 
62273 

■62 377 
62480 
62583 
62685 
62788 

.62 890 
62992 
63094 

63195 
63296 

•63 397 
63498 

63599 
63699 

63799 
■63 899 
63998 
64098 
64197 
64296 

•64 395 
64493 
64591 
64689 
64787 

.64 885 
64982 

65079 
65 176 

65273 



.60 271 
60379 
60487 

60595 
60703 

.60810 
60917 
61 023 
61 130 
61 236 

•61 342 
61 448 

61553 
61658 
61 763 

.61 868 

61 972 
62076 

62 180 
62 284 

.62 387 
62490 

62593 
62696 

62 798 
.62900 

63 003 
63104 
63205 
63306 

.63 407 
63508 
63609 
63709 
63809 

.63 909 
64008 

64 108 
64207 
64306 

.64 404 

64503 
64601 
64699 
64797 
.64 895 
64992 
65089 

65 186 
65283 



.60 282 
60390 
60498 
60606 
60713 

.60821 
60927 
61034 
61 140 
61247 

.61 352 
61458 
61563 

61 669 
61773 
.61 878 
61982 
62086 

62 190 
62294 

•62 397 
62500 
62603. 

62 706 
62808 
.62910 
63012 
63114 
63215 
63317 

•63417 
63518 
63619 

63 7«9 
63819 

•63 919 
64018 
64118 
64217 
64316 

.64414 

64513 
64611 
64709 
64807 

.64 904 
66 002 

65099 
65 196 
65292 



.60 304 
60412 
60520 
60627 
6073s 

.60842 
60949 
61055 
61 162 
61268 

•61 374 
61479 
61584 

61 690 
61794 

.61 899 

62 003 

62 107 

62 211 
62315 

.62418 
62521 
62624 
62 726 
62829 

•62931 
63033 
63134 
63236 

63337 

.63438 
63538 
63639 
63739 
63839 

.63929 .63939 
64 028 64 038 

64 128 64 137 
64227 64237 
64326 64335 

.64424 .64434 
64523 64532 
64621 64631 
64719 64729 
64816 64826 

.64914 .64924 
65011 65021 

65 108 65 118 
65205 65215 
65302 65312 



,60 293 
60401 
60509 
60617 
60724 

.60 831 
60938 
61045 
61 151 
61 257 

•61 363 
61 469 

61574 

61 679 

6i 784 

.61 888 

61993 
62097 
62201 
62304 

.62 408 
62511 
.62 613 

62 716 
62818 

.62 921 
63022 
63124 
63225 
63327 

.63 428 
63528 
63629 
63729 
63829 



LOGS. 5 PL. 
4. 



(i8) 



FIVE PLACE LOGARITHMS. 







. 5 PL. 


LOGS 


No. 
4.50 


01234 


56789 


INTP 
TAB. 




.65 321 .65 331 .65 341 .65 350 .65 360 


•65 369 65 379 .65 389 .65 398 .65 408 


10 


.6( 


•SI 


65418 65427 65437 65447 65456 


65466 65475 65485 65495 65504 


I 


.6£ 


■52 


65514 65523 65533 65543 65552 


■ 65 562 65 571 65 581 65 591 65 600 


2 




•53 


65 610 65 619 65 629 65 639 65 648 


65 658 65 667 65 677 65 686 65 696 


3 




•54 


65 706 65 715 65 725 65 734 .65 744 


65 753 65 763 65 772 65 782 65 792 


4 




4-55 


.65 8oi .65 81 1 .65 820 .65 830 .65 839 


.65 849 .65 858 .65 868 .65 877 .65 887 


5 




•56 


65896 65906 65916 65925 65935 


65944 65954 65963 65973 65982 


6 




•57 


65 992 66 001 66 01 1 66 020 66 030 


66039 66049 66058 66068 66077 


7 




•58 


66087 66096 66106 66 115 66124 


66134 66143 66153 66162 66172 


8 




•59 


66 181 66 191 66200 66210 66219 


66229 66238 66247 66257 66266 


9 




4.60 


.66276 .66285 -66295 .66304 .66314 


.66323 .66332 .66342 .66351 .66361 


9 




.61 


66370 66380 66389 66398 66408 


66417 66427 66436 66445 66455 


1 




.62 


66464 66474 66483 66492 66502 


66 511 66521 66530 66539 66549 


2 




•63 


66558 66567 66577 66586 66596 


66605 66614 66624 66633 66642 


3 




.64 


66652 66661 66671 66680 66689 


66699 66708 66717 66727 66736 


4 




4-65 


.66745 .66755 .66764 .66773 -66783 


.66792 .66801 .66811 .66820 .66829 


5 




.66 


66839 66848 66857 66867 66876 


66885 66894 66904.66913 66922 


5 




.67 


66 932 66 941 66 950 66 960 66 969 


66978 66987 66997 67 006 67015 


6 




.68 


67025 67034 67043 67052 67062 


67071 67080 67089 67099 67108 


7 




.69 


67 117 67127 67136 67145 67154 


67 164 67 173 67 182 67 191 67 201 


8 




4,70 


.67 210 .'67 219 .67 228 .67 237 .67 247 


.67256 .67265 .67274 .67284 .67293 


9 




•71 


67302 67 311 67321 67330 67339 


67348 67357 67367 67376 67385 


1 




.72 


67394 67403 67413 67422 67431 


67440 67449 67459 67468 67477 


2 




•73 


67486 67495 67504 67514 67523 


67532 67541 67550 67560 67569 


3 




•74 


67578 67587 67596 67605 67614 


67624 67633 67642 67651 67660 


4 




4^75 


.67 669 .67 679 .67 688 .67 697 .67 706 


.67 715 .67 724 .67 733 .67 742 .67 752 


5 




.76 


67 761 67 770 67 779 67 788 67 797 


67806 67815 67825 67834 67843 


5 




■77 


67852 67861 67870 67879 67888 


67897 67906 67916 67925 67934 


6 




.78 


67943 67952 67961 67970 67979 


67988 67997 68 006 68015 68024 


7 




•79 


68034 68043 68052 68061 68070 


68079 68088 68097 68106 68115 


8 




4.80 


.68 124 .68 133 .68 142 .68 151 .68 160 


.68 169 .68 178 .68 187 .68 196 .68 205 


9 




.81 


68215 68224 68233 68242 68251 


68260 68 269 68 278 68287 68296 


I 




.82 


68305 68314 68323 68332 68341 


68350 68359 68368 68377 68386 


2 




•83 


68395 68404 68413 68422 68431 


68440 68449 68458 68467 68476 


3 




.84 


68485 68494 68502 68511 68520 


68529 68538 68547 68556 68565 


4 




4.85 


.68574 .68583 .68592 .68601 .68610 


.68 619 .68 628 .68 637 .68 646 .68 655 


5 




.86 


68 664 68 673 68 681 68 690 68 699 


68708 68717 68726 68735 68744 


5 




•87 


68753 68762 68771 68780 68789 


68797 68806 68815 68824 68833 


6 




.88 


68842 68851 68860 68869 68878 


68 886 68895 68904 68913 68922 


7 




.89 


68931 68940 68949 68958 68966 


68975 68984 68993 69 002 69011 


8 




4.90 


.69020 .69028 .69037 .69046 .69055 


.69064 .69073 .69082 .69090 .69099 


8 




•91 


69108 69 117 69126 69135 69144 


69 152 69 161 69 170 69 179 69 188 


1 




.92 


69 197 69 205 69 214 69 223 69 232 


69 241 69 249 69 258 69 267 69 276 


2 




•93 


69285 69294 69302 69 311 69320 


69329 69338 69346 69355 69364 


2 




•94 


69373 69381 69390 69399 69408 


69417 69425 69434 69443 69452 


3 




4-95 


.69 461 .69 469 .69 478 .69 487 .69 496 


.69504 -69513 .69522 .69531 .69539 


4 


.60 


.96 


69548 69557 69566 69574 69583 


69 592 69 601 69 609 69 618 69 627 


S 


.69 


•97 


69 636 69 644 69 653 69 662 69 671 


69 679 69 688 69 697 69 705 69 714 


6 




.98 


69723 69732 69740 69749 69758 


69767 69775 69784 69793 69801 


6 




■99 


69 810 69 819 69 827 69 836 69 845 


69854 69862 69871 69880 69888 


7 






(19) 5 PL. I 


-OG 
4. 


S. 



5. 
LOGS. 5 PL. 



FIVE PLACE LOGARITHMS. 



No. 



8 



5.00 

.OI 
.02 

•°3 
.04 

5°5 
.06 
.07 
.08 
.09 

5.10 

.11 
.12 

•13 
.14 

S-I5 
.16 

■>7 
.18 

•19 

5.20 

.21 

.22 

•23 
.24 

5'2S 
.26 
.27 
.28 
.29 

5.30 

•31 
•32 
•33 

•34 

5^35 
•36 
•37 
•38 
■39 

5.40 

.41 
.42 
■43 
■44 

S-45 
.46 

•47 



■49 

logs! 
5. 



.69 897 
69984 
70070 
70157 
70243 
•70 329 

7041.5 
70 501 
70586 
70672 

•70 757 
70842 

70927 
71 012 
71 096 

.71 181 
71265 
71349 
71433 
71517 

.71 600 
71 684 
71767 
71850 
71933 
.72016 
72099 

72 i8i 
72263 
72346 

.72428 
72509 
72591 
72673 
72754 

•72 83s 
72916 

72997 
73078 
73159 

•73 239 
73320 
73400 
73480 

73 560 

•73 640 
73719 
73 799 
73878 
73 957 

5 PL. 



.69 go6 
69992 
70079 
70165 
70252 

•70 338 
70424 
70509 

70595 
70680 

.70 766 
70851 

70935 
71 020 
71 105 

.71 189 
71273 

71357 
71441 

71525 

,71 609 

71 692 

7177s 
71858 
71941 

.72 024 

72 107 
72 189 
72 272 
72354 

■72 436 
72518 

72599 
72681 

72 762 

.72 843 

72925 

73 006 
73086 
73167 

■73 247 
73328 
73408 
73488 
73568 

•73 648 
73727 
73807 
73886 

73965 



.69914 
70 001 

70088 
70174 
70 260 

.70 346 
70432 
70518 
70603 
70689 

■70 774 

70 859- 
70944 

71 029 
71 "3 
.71 198 
71 282 
71366 
71450 
71533 

.71617 
71 700 
71784 

71 867 
71950 
.72032 

72115 

72 198 
72 280 
72362 

.72444 
72 526 
72607 
72689 

72 770 

.72852 
72933 
73014 
73094 
73175 

■73255 
73336 
73416 
73496 
73576 

•73656 

73 735 
73 815 
73894 
73 973 



,69 923 
70010 
70096 
70183 
70269 

•70355 
70441 

70 526 
70612 
70697 

•70 783 
70868 
70952 
71037 

71 122 

.71 206 
71 290 
71374 
71458 
71542 

.71 625 

71 709 
71792 
71875 
71958 

.72 041 

72 123 
72206 
72288 
72370 

•72452 
72534 
72616 
72697 
72779 
,72860 
72941 
73022 
73102 
73183 

•73 263 

73 344 
73424 
73504 
73584 

■73 664 
73 743 
73823 
73902 
73981 



.69 932 
,70018 
70 105 
70191 
70 278 

■70 364 

70449 

70535 
70621 

70 706 

•70 791 
70876 
70961 

71 046 
71 130 

.71 214 
71299 

71383 
71 466 

71550 

•71 634 
71717 
71 800 
71883 

71 966 

.72049 
72132 

72 214 
72296 
72378 

.72 460 
72542 
72624 
72705 

72 787 
.72 868 
72949 
73030 

73 III 
73 191 

■73 272 
73352 
73432 
73512 
73 592 
■73672 

73751 
73830 
73910 
73989 



.69 940 
70027 
70 1 14 

70 200 
70286 

•70372 
70458 
70544 
70629 
70714 

.70800 
70885 
70969 
71054 

71 139 
.71 223 

71307 
71 391 
71475 
71559 

.71 642 
71725 
71 809 

71 892 
71975 
.72057 

72 140 

72 222 
72304 
72387 

•72 469 
72550 
72632 

72713 
72795 

.72876 
72957 
73038 

73 "9 
73199 

.73 280 
73360 
73440 
73520 
73600 

•73 679 
73 759 



73918 
73 997 



,69 949 
70036 
70 122 

70 209 
70295 

.70 381 
70467 

70552 
70638 

70723 

.70 S08 
70893 
70978 
71063 

71 147 

.71 231 

71315 
71399 
71483 
71567 

.71 650 

71734 
71817 

71 900 
71983 
.72066 

72 148 
72230 
72313 
72395 

.72477 

72558 
72 640 
72 722 
72803 

.72 884 
72965 
73046 
73127 
73207 

.73 288 
73368 
73448 
73528 
73608 

■73687 
Z376Z 

73926 
74 005 



.69958 , 
70044 
70131 
70217 
70303 
■70 389 
70475 
70561 
70646 
70731 

.70817 
70902 
70986 
71 071 
71155 
.71 240 

71324 
71 408 
71492 
71575 

.71 659 
71742 
71825 
71 908 
71991 

.72074 
72156 

72239 
72321 

72403 

.72485 
72567 
72648 
72730 
72811 

.72 892 
72973 
73054 
73135 
73215 

•73 296 
73376 
73456 
73536 
73616 

•73 695 
73 775 
73854 
73 933 
74013 



,69 966 
70053 

70 140 
70226 
70312 

■70 398 
70484 
70569 
70655 
70740 

.70 825 
70910 
70995 
71079 

71 164 

.71 248 
71332 
71416 

71 500 
71584 

.71 667 
71750 

71834 
71917 
71999 

.72 082 

72 165 
72247 
72329 
72411 

■72493 
72575 
72656 

72738 
72819 

.72900 
72981 
73062 

73143 
73223 

■73 304 
73384 
73464 

73 544 
73624 

■73 703 
73783 
73862 

73941 
74020 



■69 975 

70 062 
70148 
70234 
70321 

.70 406 
70492 
70578 
70663 
70749 

.70834 
70919 
71003 
71088 
71172 

•71 257 
71341 
71425 
71508 
71592 

•71 675 
71759 

71 842 

71925 

72 008 

.72 090 
72173 
72255 
72337 
72419 

•72 501 
72583 
72665 
72746 
72827 

.72 908 

72989 
73070 

73151 
73231 

•73312 
73392 
73472 
73552 
73632 

■73711' 

73791 

73870 

73 949 
74028 



(20) 



FIVE PLACE LOGARITHMS. 



5 PL. LOGS. 



No. 



8 



5.50 

•51 
•52 

•53 
•54 

5^55 
.56 

■57 
•58 
•59 

5.60 

.61 
.62 

•63 
.64 

5-65 
.66 

.67 
.68 
.69 

5.70 

•7> 

.72 

•73 
•74 

5-75 
.76 

•77 
.78 

•79 

5.80 

.81 
.82 

•83 
.84 

5.85 
.86 
.87 



5.90 

•91 
.92 

•93 
•94 

5-95 
.96 

•97 
:98 

•99 



.74036 

74 "5 
74194 

74 273 
74351 

•74 429 
74507 
74586 
74663 
74741 

.74819 
74896 
74974 
75051 
75128 

■75 205 
75282 
75358 

75 435 

75 5" 

•75 587 
75664 

75740 
75815 
75891 

•75 967 
76042 

76 118 

76193 
76268 

■76 343 
76418 
76492 
76567 
76641 

.76 716 

76 790 
76864 

76938 
77012 

•77085 

77159 
77232 

77305 

77 379 
•77452 

77525 
77 597 
77 670 

77 743 



.74044 

74123 
74202 
74280 
74 359 

•74437 
745'5 
74 593 
74671 

74 749 

•74 827 
74904 
74981 
75059 
75136 

•75213 
75289 

75366 
75442 
75519 

■75 595 
75671 

75 747 
75823 
75899 

■75 974 
76050 
76125 
76200 
76275 

•76 350 
76425 
76500 

76574 
76649 

•76 723 

76797 
76871 

76945 
77019 

•77 093 
77 166 
77 240 

77313 
77386 

•77 459 
77532 
77605 
77677 
77750 



•74 052 
7413J 

74 2IO 
74288 
74367 

•74 445 
74523 
74601 
74679 

74 757 

•74 S34 
74912 
74989 
75066 
75143 

.75 220 

75297 

75 374 
75450 
75526 

•75 603 
75679 

75 755 
75831 
75906 

•75 982 
76057 

76133 
76208 
76283 

•76358 
76433 
76507 
76582 
76656 

.76 730 
76805 

76 879 
76953 
77026 

.77 100 

77173 
77247 
77320 

77 393 
.77 466 

77 539 
77 612 
77685 
77 757 



.74 060 

74139 
74218 
74296 
74 374 

■74 453 
74 531 
74609 
74687 
74764 

.74 842 
74920 

74 997 
75074 
75151 

.75 228 
75305 
75381 
75458 

75 534 

.75 610 
75686 
75762 
75838 
75914 

•75 989 
76065 

76 140 
76215 
76290 

•76365 
76440 

76515, 

76589 

76664 

•76 738 
76812 
76886 
76960 
77034 

•77 107 

77 181 

77254 
77327 
77401 

•77 474 
77546 
77619 
77 692 
77764 



.74 068 

74147 
74225 

74304 
74382 

.74461 

74 539 
74617 
74695 
74772 

■74 850 
74927 

75 005 

75 082 
75159 

■75 236 
75312 
75389 
75465 
75542 

.75618 
75694 
75770 
75846 
75921 

■75 997 
76072 
76148 
76223 

76 298 

•76 373 
76448 
76522 

76597 
76671 

■76 745 
76819 

76893 
76967 
77041 

■77 "5 
77188 

77 262 

77 335 
77408 

■77481 
77 554 
77 627 
77699 
77772 



.74076 
74155 
74233 
74312 
74390 

.74 468 

74 547 
74624 
74702 
74780 

.74858 

74 935 
75012 
75089 

75 i66 

•75 243 
75320 
75 397 
75 473 
75 549 

■75 626 
75702 
75778 
75853 
75929 
.76 005 
76080 
76155 
76230 
76305 

.76 380 

76455 
76530 
76604 
76678 

■76753 
76827 
76901 

7697s 
77048 

■77 122 
77 195 
77269 
77342 
77415 

■77 488 
77561 

77634 
77 706 

77 779 



.74084 
74162 

74241 
74320 
74398 

■74476 
74 554 
74632 
74710 
74788 

•74865 

74 943 

75 020 

75 097 
75174 

•75251 
75328 
75404 
75481 

75 557 

•75 633 
75709 
75785 
75861 

75 937 
.76012 
76087 

76 163 
76238 
76313 

.76388 
76462 

76537 
76612 
76686 

.76 760 

76834 
76908 
76982 
77056 

■77 129 
77203 

77 276 

77 349 
77422 

■77 495 
77568 
77641 

77714 
77786 



.74092 
74170 
74249 
74327 
74406 

•74 484 
74562 

74 640 
74718 
74796 

•74873 
74950 
75028 
75105 
75182 

•75 259 

75 335 
75412 

75488 

75565 

•75 641 
75717 
75 793 
75868 

75 944 
.76 020 
76095 

76 170 
76245 
76320 

•76395 
76470 

76545 
76619 
76693 

.76 768 
76842 
76916 
76989 
77063 

■77137 

77 210 

77283 
77 357 
77430 

•77 503 
77576 
77648 
77721 
77 793 



74099 .74107 
74 178 74 186 
74257 74265 
74 335 74 343 
74414 74421 

74492 .74500 
74570 74578 
74648 74656 
74726 74733 
74803 74 81 1 



.74 881 
74958 
75035 
75 "3 
75189 

•75 266 
75 343 
75420 

75496 
75572 

•75 648 
75724 

75 800 
75876 
75952 

.76027 

76 103 
76178 
76253 
76328 

.76403 
76477 
76552 
76626 

76 701 

•76 775 
76849 
76923 

76997 
77070 

■77 144 
77217 
77291 

77364 

77 437 



.74 889 

74966 

75043 
75120 

75197 
•75 274 
75351 
75427 
75504 
75580 

■75 656 
75732 
75808 
75884 

75 959 

■76 035 

76 110 
76185 
76 260 
7633s 

.76410 
76485 
76559 
76634 

76 708 

.76 782 
76856 
76930 

77 004 
77078 

■77 151 
77225 
77298 

77371 
77 444 



.77510 .77517 

77583 77590 

77656 77663 

77728 77735 

77 801 77 808 



INTP. 
TAB. 



(21) 



5 PL. LOGS. 
5. 



FIVE PLACE LOGARITHMS. 



LOGS. 


5 PL. 








No. 


01234 


56789 


INTP. 
TAB. 


.77 
.84 


6.00 

.OI 
.02 

•03 
.04 


.77815 .77822 .77830 .77837 .77844 
77887 77895 77902 77909 77916 
77960 77967 77974 77981 77988 
78032 78039 78046 78053 78061 
78104 78 III 78118 78125 78132 


.77851 .77859 -77866 .77873 .77880 
77924 77931 77938 77945 77952 
77996 78 003 78010 78017 78025 
78068 78075 78082 78089 78097 
78 140 78 147 78 154 78 161 78 168 


8 
I 

2 
2 
3 




6.05 
.06 
.07 
.08 
.09 


.78 176 .78 183 .78 190 .78 197 .78204 
78247 78254 78262 78269 78276 
78319 78326 78333 78340 78347 
78390 78398 78405 78412 78419; 
78462 78469 78476 78483 78490 


.78211 .78219 .78226 .78233 .78240 
78283 78290 78297 78305 78312 
78355 78362 78369 78376 78383 
78426 78433 78440 78447.78455 
78497 78504 78512 78519 78526 


4 
5 
6 

6 
7 




6.10 

.11 
.12 

•13 
.14 


-78533 -78540 -78547 -78554 -78561 
78604 78611 78618 78625 78633 
78675 78682 78689 78696 78704 
78746 78753 78760 78767 78774 
78817 78824 78831 78838 78845 


.78569 .78576 .78583 .78590 .78597 
78640 78647 78654 78661 78668 
78711 78718 78725 78732 78739 
78781 78789 78796 78803 78810 
78852 78859 78866 78873 78880 


7 

1 
1 
2 
3 




6.15 
.16 

■17 
.18 
.19 


-78888 -78895 .78902 .78909 .78916 
78958 78965 78972 78979 78986 
79029 79036 79043 79050 79057 
79099 79106 79113 79120 79127 
79,169 79176 79183 79190 79197 


.78923 .78930 .78937 .78944 .78951 
78993 79 000 79007 79014 79021 
79064 79071 79078^79085 79092 
79134 79141 79148 79155 79162 
79204 79 2U 79218 79225 79232 


4 
4 
5 
6 
6 




6.20 

.21 
.22 

•23 
.24 


-79 239 -79 246 -79 253 .79 260 .79 267 
79309 79316 79323 79330 79 337 
79 379 79386 79 393 79 400 79407 
79 449 79456 79463 79470 79 477 
79518 79525 79532 79 539 79546 


•79 274 -79 281 .79 288 ..79 295 .79 302 
79 344 79351 79358 79365 79372 
79414 79421 79428 79435 79442 
79484 79491 79498 79505 79 511 
79 553 79560 79567 79 574 79 581 


7 

1 
I 
2 
3 




6.25 
.26 
.27 
.28 
.29 


.79 588 .79 595 .79 602 .79 609 .79 616 
79657 79664 79671 79678 79685 
79727 79 734 79741 79748 79 754 
79 796 79 803 79 810 79 817 79 824 
79865 79872 79879 79886 79893 


•79 623 .79 630 .79 637 .79 644 .79 650 
79692 79699 79706 79713 79720 
79761 79768 79775 79782 79789 
79831 79837 79844 79851 79858 
79900 79906 79913 79920 79927 


4 
4 
5 
6 
6 




6.30 

■31 
•32 
•33 
•34 


•79 934 -79 941 .79948 -79 955 -79962 
80 003 80010 80017 80024 80030 
80072 80079 80085 80092 80099 
80 140 80 147 80 154 80 161 80 168 
80209 80216 80223 80229 80236 


•79 969 ^79 975 ^79 982 .79 989 -79 996 
80037 80044 80051 80058 80065 
80106 80113 80120 80127 80134 
80175 80182 80188 80195 80202 
80 243 80 250 80 257 80 264 80 271 


7 

1 
1 
2 
3 




6.35 
-36 
•37 
•38 
•39 


.80 277 .80 284 .80 291 .80 298 .80 305 
80346 80353 80359 80366 80373 
80414 80421 80428 80434 80441 
80 482 80 489 80 496 80 502 80 509 
80550 80557 80564 80570 80577 


.80312 .80318 .80325 .80332 .80339 
80380 80387 80393 80400 80407 
80448 80455 80462 80468 80475 
80516 80523 80530 80536 80543 
80584 80591 80598 80604 80611 


4 
4 
5 
6 
6 




6.40 

■41 
.42 

•43 
•44 


.80618 .80625 -80632 .80638 .80645 
80686 80693 80699 80706 80713 
80 754 80 760 80 767 80 774 80 781 
80821 80828 80835 80841 80848 
80889 80895 80902 80909 80916 


.80652 .80659 .80665 -80672 .80679 
80 720 80 726 80 733 80 740 80 747 
80787 80794 80801 80808 80814 
80855 80862 80868 80875 80882 
80 922 80 929 80 936 80 943 80 949 


6 

I 
1 
2 
2 


.77 
.84 


6-45 
.46 

-47 
.48 
-49 


.80956 .80963 .80969 .80976 .80983 
81023 81030 81037 81043 81050 
81090 81097 81104 81111 81117 
81 158 81 164 81 171 81 178 81 184 
81224 81231 81238 81245 81251 


.80990 .80996 .81003 .81010 .81017 
81 057 81 064 81 070 81 077 81 084 
81 124 81 131 8i 137 81 144 81 151 
81191 81198 81204 81 2U 81218 
81 258 81 265 81 271 81 278 81 285 


3 

4 
4 
5 
5 


LO 

e 


GS. 


5 PL. (22) 







FIVE PLACE LOGARITHMS. 













6 PL. 


LOGS 


No. 


12 


3 4 


5 


6 


789 


INTP. 
TAB. 

6 




6.50 


.81 291 .81 298 .81 305 


.81311 .81318 


.81 325 


•81 331 


.81338 .81 345 -81351 


.7' 


•5' 


81 358 81 365 81 371 


81378 81385 


81 391 


81398 


81405 81411 81418 


1 


.84 


•52 


81425 81 431 81438 


81445 '81451 


81458 


81465 


81471 81478 81485 


I 




•53 


81 491 81 498 81 505 


81 511 81 518 


81525 


81 531 


81538 81544 81-551 


2 




•54 


81 558 81564 81 571 


81 578 81 584 


81591 


81598 


81 604 81 611 81617 


2 




6.55 


.8i 624 .81 631 .81 637 


.81 644 .81 651 


■81 657 


.81 664 


.81 671 .81 677 .81 684 


3 




.56 


81 690 8i 697 81 704 


81 710 81 717 


81723 


81730 


81 737 81 743 81 750 


4 




■57 


81 757 81 763 81 770 


81 776 81 783 


81790 


81796 


81 803 81 809 81 816 


4 




.58 


81 823 81 829 81 836 


81 842 81 849 - 


81856 


81862 


81 869 81 875 81 882 


5 




•59 


81 889 81 895 81 902 


81 908 81 915 


81 921 


81928 


81 935 81 941 81 948 


5 




6.60 


.81 954 .81 961 .81 968 


.81 974 .81 981 


.81 987 


.81 994 


.82 000 .82007 -82014 


% 




.61 


82020 82027 82033 


82 040 82 046 


82053 


82060 


82 066 82 073 82 079 


I 




.62 
•63 


82 086 82 092 82 099 
82 151 82158 82164 


82105 82 112 
82 171 82178 


82 119 
82184 


82 12£ 


82 132 82 138 82 145 


I 
2 




8219^1 


82 197 82 204 82 210 




.64 


82217 82223 82230 


82 236 82 243 


82 249 


82256 


82263-82269 82276 


3. 




6.65 


.82282 .82289 .82^295 


.82 302 .82 308 


.82315 


.82321 


.82328 .82334 .82341 


4 




.66 


82347 82354 82360 


82367 82373 


82380 


82387 


82 393 82 400 82 406 


4 




.67 


82413 82419 82426 


82432 82439 


82445 


82452 


82458 82465 82471 


5 




.68 


82478 82484 82491 


82 497 82 504 


82 510 


82517 


82523 82530 82536 


6 




.69 


82543 82549 82556 


82 562 82 569 


82575 


82 582 


82588 82595 82601' 


6 




6.70 


.82 607 .82 614 .82 620 


.82627 .82633 


.82 640 


.82 646 


.82 653 .82 659* .82 666 


6 




•71 


82672 82679 82685 


82692 82698 


82705 


82 71 I 


82 718 82 724 82 730 


I 




.72 


82737 82743 82750 


82756 82763 


82 769 


82776 


82782 82789 82795 


I 




■73 


82802 82808 82814 


82821 82827 


82834 


82840 


82847 82853 82860 


2 




•74 


82866 82872 82879 


82885 82892 


82898 


82 905 


82911 82918 82924 


2 




6.75 


.'82 930 .82 937 .82 943 


.82950 .82956 


■82 963 


.82 969 


.82975 .82982 .82988 


3 




.76 


82995 83 001 83008 


83 014 83 020 


' 83027 


83033 


83040 83046 83052 


4 




•77 


83059 83065 83072 


83078 83085 


83091 


83097 


83104 83110 83117 


4 




.78 


83 123 83 129 83 136 


83 142 83 149 


83155 


83 161 


83 168 83 174 83 181 


5 




•79 


83 187 83 193 83 200 


83 206 83 213 


83219 


83225 


83232 83238 83245 


5 




6.80 


.83251 .83257 .83264 


.83270 .83276 


•83 283 


.83 289 


.83 296 .83 302 .83 308 


7 




.81 


83315 83321 83327 


83334 83340 


83347 


83353 


83359 83366 83372 


1 




.82 


83"378 83385 83391 


83398 83404 


83410 


83417 


83423 83429 83436 


1 




•83 


83442 83448 83455 


83461 83467 


83474 


83480 


83487 83493 83499 


2 




.84 


83506 83512 83518 


83525 83531 


83537 


83544 


83550 83556 83563 


3 




6.85 


■83 569 ^83 575 •SS 582 


•83 588 .83 594 


.83 601 


.83 607 


.83613 .83620 .83626 


4 




.86 


83632 83639 83645 


83651 83658 


83664 


83670 


83677 83683 83689 


,4 




.87 


83696 83702 83708 


83715 83721 


83727 


83734 


83740 83746 83753 


5 




.88 


83759 83765 83771 


83778 83784 


83790 


83797 


83803 83809 83816 


6 




.89 


83822 83828 83835 


83841 83847 


83853 


83860 


83866 83872 83879 


6 




6.90 


.83885 .83891 .83897 


.83904 .83910 


.83916 


•83 923 


-83 929 -83 935 -83 942 


6 




•91 


83948 83954 83960 


83967 83^973 


83979 


83985 


83992 83998 84 004 


1 




.92 


84 on 84017 84023 


84 029 84 036 


84042 


84048 


84 055 84 061 84 067 


1 




■93 


84073 84080 84086 


84 092 84 098 


84105 


84 III 


84 117 84123 84130 


2 . 




■94 


84136 84142 84148 


84155 84 161 


84167 


84173 


84180 84186 84192 


2 




6.95 


.84198 .84205 .84211 


.84217 .84 223 


.84 230 


.84 236 


.84 242 .84 248 .84 255 


3 




.96 


84 261 84 267 84 273 


84280 84286 


84 292 


84298 


84305 84311 84317 


4 


.Ti 


•97 


84323 84330 84336 


84342 84348 


84354 


84361 


84367 84373 84379 


4 


.84 


.98 


84386 84392 84398 


84404 84410 


84417 


84423 


84429 84435 84442 


5 




■99 


84448 84454 84460 


84 466 84 473 


84479 


84485 


84491 84497 84504 


5 




^ 




(23) 




6 PL. 


-OC 
6. 


s 



7. 
LOGS. 6 PL. 



FIVE PLACE LOGARITHMS. 



No. 



7.00 

.OI 
.02 

•03 

.04 

7-05 
.06 
.07 
.08 
.09 

7.10 

.II 

.12 

•13 
.14 

7-"5 
.16 

•17 
.18 
.19 

7.20 

.21 
.22 

•23 

.24 

7-25 
.26 

•27 
.28 
.29 

7.30 

•31 
•32 
■33 
•34 

7-35 
•36 
•37 
•38 
•39 

7.40 

.41 
•42 
•43 
■44 

7-45 
.46 

•47 
.48 

•49 



LOGS. 
7, 



8 



.84 510 .84516 
S4572 84578 
84 634 84 640 
84 696 84 702 
84757 84763 

.84819 .84825 
84880 84887 

84 942 84 948 

85 003 85009 
85065 85071 



.85 126 
85187 
85248 
85309 
85370 

•85 431 
85491 

85552 
85612 

85673 

•85 733 
85794 
85854 
85914 
85974 

.86 034 
86094 
86153 
86213 
86273 

•86 332 
86392 
86451 
86510 
86570 

.86 629 
86 688 

86747 
86806 
86864 

.86 923 
86982 
87040 
87099 
87157 

.87216 
87274 
87332 
87390 
87448 

5 PL. 



.85 132 
85193 
85254 
85315 
85378 

85 437 
85497 
85558 
85618 
85679 

■85 739 
85800 
85860 
85 920 

85 980 
,86 040 

86 100 
86159 
86219 
86 279 

,86 338 
86398 
86457 
86516 
86576 

,86 635 
86694 

86753 
86812 
86870 

.86 929 
86988 
87046 
87105 
87163 

.87 221 
87280 
87338 
87396 
87454 



.84 522 

84584 
84646 
84708 
84770 

.84 831 
84893 
84954 
85016 
85077 

.85 138 

85199 
85260 

85321 
85382 

■85443 
85503 
85564 
85625 
85685 

•85 745 
85806 
85866 
85926 
85986 

.86 046 
86106 
86165 
86225 
86285 

.86 344 
86404 
86463 
86522 
86581 

.86 641 

86 700 

86759 
86817 
86876 

•86935 
86994 
87052 

87 III 
87169 

.87 227 
87286 

87344 
87402 
87460 



.84 528 
84590 
84652 
84714 
84776 

.84837 
84899 
84960 

85 022 
85083 

•85 144 
85205 
85266 

85327 
85388 

■85449 
85509 
85570 
85631 
85691 

.85751 
85812 
85.872 
85932 
85992 

.86052 

86 112 

86 171 
86231 
86291 

.86 350 
86410 
86469 
86528 
86587 

.86 646 
86705 
86764 
86823 
86882 

.86 941 
86999 
87058 

87 116 

8717s 

•87 233 
87291 

87349 
87408 
87466 



.84 535 
84597 
84658 

84 720 
84782 

.84844 

84905 
84967 
85028 
85089 

.85 150 

85 211 

85 272 
85333 
85394 

•85 455 
85516 
85576 

85637 
85697 

•85 757 
85818 
85878 
85938 
85998 

.86058 

86 118 
86177 
86237 
86 297 

.86356 
86415 

86475 
86534 
86593 
.86 652 
86 711 
86 770 
86829 

86 888 

•S6947 

87 005 

87064 
87 122 
87 181 

•87 239 
.87 297 

87355 
87413 
87471 

(24) 



.84 541 
84603 
84665 
84 726 
84788 

.84850 
84 91 1 
84973 
85034 
85095 

.85 156 
85217 
85278 

85339 
85400 

.85 461 
85522 
85582 
85643 
85703 

•85 763 
85824 
85884 

85944 
86 004 

.86 064 

86 124 
86183 
86243 
86303 

.86 362 
86421 
86481 
86540 
86599 

.86 658 
86717 
86776 
86835 
86894 

■86953 

87 01 1 
87070 
87 128 
87186 

■87 245 
87303 
87361 
87419 
87477 



,84 547 
84609 
84671 

84733 
84794 

.84 856 
84917 
84979 
85 040 
85 101 

.85 163 
85 224 
85285 

85345 
85406 

.85 467 
85528 
85588 
85649 
85709 

.85 769 
85 830 
85890 
85950 
86010 



■84553 
84615 
84677 

84739 
84800 

.84 862 
84924 

84985 
85046 
85107 

.85 169 
85 230 
85291 

85352 
85 412 

•85 473 
85534 
85594 
85655 
8571S 

•85 775 
85836 
85896 
85956 
86016 



•84 559 
84621 
84683 

84745 
84807 

.84 868 
84930 
84991 
85 052 
85114 

•85 17s 
85236 

85297 
85358 
85418 

•85 479 
85540 
85 600 
85661 
85721 



.84 566 
84628 
84689 

84751 
84813 

.84 874 
84936 

84997 
85058 
85 120 

.85 181 
85 242 
85303 
85364 
85425 

•85 485 
85546 
85606 
85667 
85727 



.86070 .86076 
86130 86136 
86 189 86 195 
86 249 86 255 
86308 86314 



.85 781 .85 788 
85842 85848 

85 902 85 908 
85962 85968 
86022 86028 

.86082 .86088 

86 141 86 147 
86 201 86 207 
86261 86267 
86320 86326 



.86 368 
86427 
86487 
86546 
86605 

.86 664 
86723 
86782 
86841 
86900 

.86 958 
87017 
87075 
87134 
87192 

.87 251 
87309 
87367 
87425 

87483 



.86374 
86433 
86493 
86552 

86 61 1 

.86 670 
86729 
86788 
86847 
86906 

.86 964 
87023 
87081 

87 140 
87198 

.87 256 

87315 

87373- 

87431 

87489 



,86 380 
86439 
86499 
86558 
86617 

.86 676 

86735 
86794 
86853 
86 91 1 

,86 970 
87029 
87087 
87146 
87204 

.87 262 
87320 
87379 
87437 
87495 



,86 386 
86445 
86504 
86564 
86623 

,86 682 
86741 
86800 
86859 
86917 

.86976 

87035 
87093 
87 151 
87 210 

.87 268 
87326 

87384 
87442 
87 500 



FIVE PLACE LOGARITHMS. 



6 PL. LOGS. 



No. 


I 


2 


3 


4 


5 


6 


789 


INTR 
TAB. 


7.50 


.87 506 .87 512 


.87518 


•87 523 


.87 529 


•87 535 


•87 541 


■87547 -87552 .87558 


6 


•5' 


87564 87570 


87576 


87581 


87587 


87593 


87599 


87 604 87 610 87 616 


1 


•52 


87622 8762S 


87633 


87639 


87645 


87651 


87656 


87662 87668 87674 


1 


•S3 


87.679 87685 


87691 


87697 


87703 


87708 


87714 


87 720 87 726 87 731 


2 


•54 


87737 87743 


87749 


87754 


87760 


87766 


87772 


87777 87783 87789 


2 


7^55 


.87 795 .87 800 


.87 806 


.87812 


.87818 


■87 823 


.87 829 


.87835 .87841 .87846 


3 


.56 


87852 87858 


87864 


87869 


87875 


87881 


87887 


87892 87898 87904 


4 


•57 


87910 87915 


87921 


87927 


87933 


87938 


87944 


87950 87955 87961 


4 


■58 


87967 87973 


87978 


87984 


87990 


87996 


88 001 


88007 88013 88018 


5 


•59 


88024 88030 


88036 


88041 


88047 


88053 


88058 


88064 88070 88076 


5 


7.60 


.88081 .88087 


.88093 


.88 098 


.88 104 


.88110 


.88116 


.88 121 .88 127 .88 133 


S 


.61 


88138 88144 


88150 


88156 


88161 


88167 


88173 


88 178 88 184 88 190 


1 


.62 


88195 88201 


88207 


88213 


88218 


88224 


88230 


88235 88241 88247 


1 


•63 


88252 88258 


88264 


88270 


88275 


88281 


88287 


88292 88298 88304 


2 


.64 


88309 88315 


88321 


88326 


88332 


88338 


88343 


88349 88355 88360 


2 


7-65 


.88 366 .88 372 


■88 377 


.88 383 


.88 389 


■88 395 


.88 400 


.88406 .88412 .88417 


3 


.66 


88423 88429 


88434 


88440 


88446 


88451 


88457 


88463 88468 88474 


3 


.67 


88480 88485 


88491 


88497 


88502 


88508 


88513 


88519 88525 88530 


4 


.68 


88536 88542 


88547 


88553 


88559 


88564 


88570 


88576 88581 88587 


4 


.69 


88593 88598 


88604 


88610 


88615 


88621 


88627 


88632 88638 88-.643 


5 


7.70 


.88649 .88^55 


.88 660 


.88 666 


.88 672 


.88 677 


!88 683 


.88 689 .88 694 .88 700 


6 


•71 


88705 88 711 


88717 


88722 


88728 


88734 


88739 


88745 88750 88756 


1 


.72 


88762 88767 


88773 


88779 


88784 


88790 


88795 


88801 88807 88812 


I 


•73 


88818 88824 


88829 


88835 


88840 


88846 


88852 


88857 88863 88 868 


2 


•74 


88874 88880 


88885 


88891 


88897 


88902 


88908 


88913 88919 88925 


2 


7-75 


.88930 .88936 


.88 941 


.88 947 


.88953 


.88 958 


.88 964 


.88969 .88975 .88981 


3 


.76 


88986 88992 


88997 


89 003 


89009 


89014 


89020 


89025 89031 89037 


4 


•77 


89 042 89 048 


89053 


89059 


89064 


89070 


89076 


89081 89087 89092 


4 


.78 


89 098 89 104 


89 109 


89 115 


89 120 


89126 


89131 


89 137 89 143 89 148 


5 


•79 


89154 89159 


89165 


89 170 


89176 


89182 


89187 


89 193 89 198 89 204 


5 


7.80 


.89209 .89215 


.89 221 


.89 226 


.89 232 


.89 237 


.89 243 


.89248 .89254 .89260 


5 


.81 


89 265 89 271 


89276 


89282 


89287 


89293 


89298 


89304 89310 89315 


1 


.82 


89 321 89 326 


89.332 


89337 


89343 


89348 


89354 


89360 89365 89371 


1 


•83 


89376 89382 


89387 


89393 


89398 


89404 


89409 


89415 89421 89426 


2 


.84 


89432 89437 


89443 


89448 


89454 


89459 


89465 


89470 89476 89481 


2 


7-85 


.89 487 .89 492 


.89498 


.89 504 


.89 509 


•89515 


.89 520 


.89 526 .89 531 .89 537 


3 


.86 


89542 89548 


89553 


89559 


89564 


89570 


89575 


89581 89586 89592 


3 


•87 


89597 89 6»3 


89609 


89614 


89 620 


89625 


89631 


89 636 89 642 89 647 


4 


.88 


89653 89658 


89664 


89669 


89675 


89680 


89686 


89 691 89 697 89 702 


4 


.89 


89 708 89 713 


89719 


89724 


89730 


89735 


89741 


89746 89752 89757 


5 


7.90 


.89 763 .89 768 


■89 774 


.89 779 


•89 785 


•89 790 


■89 796 


.89 801 .89 807 .89 812 


6 


•91 


89818 89823 


89829 


89834 


89840 


89845 


89851 


89856 89862 89867 


1 


.92 


89873 89878 


89883 


89889 


89894 


89900 


89905 


89911 89916 89922 


1 


•93 


89927 89933 


89938 


89944 


89949 


89955 


89 960 


89966 89971 89977 


2 


•94 


89982 89988 


89993 


89998 


90 00i 


90009 


90015 


90020 90026 90031 


2 


7^95 


.90037 .90042 


.90 048 


.90053 


.90059 


.90 064 


.90 069 


.90075 .90080 .90086 


3 


.96 


90091 90097 


90 102 


90 108 


90 113 


90119 


90 124 


90 129 90 135 90 140 


4 


•97 


90 146 90 151 


90157 


90 162 


90 168 


90173 


90179 


90 184 90 189 90 195 


4 


.98 


90 200 90 206 


90 21 1 


90 217 


90222 


90 227 


90233 


90 238 90 244 90 249 


5 


■99 


90 255 90 260 


90266 


90271 


90 276 


90 282 


90287 


90 293 90 298 90 304 


5 



(25) 



5 PL. 



LOGS. 
7. 



8. 
LOGS. 5 PL. 



FIVE PLACE LOGARITHMS. 



No. 



8.00 

.OI 
.02 

•03 

.04 

8.05 
.06 
.07 
.08 
.09 

8.10 

.II 

.12 

•13 

.14 

8.15 
.16 

■17 
.18 

•19 

8.20 

.21 
.22 

•23 
.24 

8.25 
.26 

•27 
.28 
.29 

8.30 

•31 
•32 
■33 

•34 

8^35 
•36 
•37 
•38 
•39 

8.40 

.41 
.42 
•43 
■44 

8.45 
.46 

•47 
.48 

•49 



2 



8 



.90 309 
90363 
90417 
90472 
90526 

.90 580 
90634 
90687 
90741 
90795 

.90 849 
90902 
90956 
91 009 
91 062 

.91 116 
91 169 
91 222 
91275 
91328 

•91 381 
91434 
91487 

91 540 
91593 
.91 645 
916518 

91 751 
91803 
91855 

.91 908 

91 960 
92012 

92 065 
92 117 

.92 169 
92 221 
92273 
92324 
92376 

.92428 
92480 
92531 
92583 
92634 

.92686 . 

92737 
92788 
92 840 
92891 



.90314 
90369 
90423 
90477 
90531 
.90 585 
90639 
90693 
90747 
90800 

•90 854 
90907 
90961 
91 014 
91 068 

.91 121 

91 174 
91 228 
91 281 
91334 

•91' 387 
91440 
91492 

91545 
91598 

.91 651 

91703 
91756 
91 808 

91 861 

•91 913 
91965 
92018 
92070 

92 122 

•92 174 
92 226 
92 278 
92330 
92381 

•92 433 
92485 
92536 
92588 
92639 

.92 691 
92742 
92793 
92845 
92896 



.90 320 

90374 
90428 
90482 
90536 
.90 590 

90 644 
90698 
90752 
90806 

•90 859 
90913 
90966 

91 020 
91073 
.91 126 
91 180 

91233 
91286 

9>339 

.91 392 

91445 
91498 

91 551 
91603 

.91 656 
91709 
91 761 
91 814 
91866 

.91 918 

91 971 
92023 
92075 

92 127 

■92179 
92231 
92283 
92335 
92387 

•92 438 
92490 
92542 

92593 
92645 

.92 696 
92747 
92799 
92 850 
92901 



.90 325 
90380 

90434 
90488 
90542 

■90 596 
90650 
90703 
90757 

90 811 

.90 865 
90918 
90972 

91 025 
91 078 
.91 132 
91 185 
91238 
91 291 
91344 

■91 397 
91450 

91503 
91556 
91 609 

.91 661 

91714 
91 766 
91 819 

91 871 

.91 924 
91976 
92028 
92080 

92 132 
.92 184 
92236 
92288 
92340 
92392 

•92 443 
92495 
92547 
92598 
92650 

.92 701 
92752 
92804 
92855 
92906 



•90 331 
90385 
90439 
90493 
90547 
.90 601 
90655 

90 709 
90763 
90816 

.90 870 
90924 

90977 
91030 

91 084 

•91 «37 
91 190 
91243 
91297 
91350 

.91 402 

91455 
91 508 
91 561 
91 614 

.91' 666 
91 719 
91 772 
91 824 
91 876 

.91 929 

91 981 
92033 
92085 
92137 
.92 189 

92 241 
92293 
92345 
92397 

•92449 
92500 

92552 
92 603 
92655 
.92 706 

92758 
92 809 
92860 
92911 



•90 336 
90390 

90445 
90499 
90553 
.90 607 
90660 
90714 
90768 
90822 

.90 875 
90929 
90982 
91036 
91 089 
.91 142 
91 196 
91249 
91302 
9135s 

.91 408 
91 461 
91 514 
91 566 
91 619 

.91 672 
91 724 
91 777 
91 829 
91 882 

■91 934 

91 986 
92038 
92091 
92143 
.92 195 .92 200 
92247 92252 
92298 92304 
92350 92355 

92 402 92 407 



,90 342 
90396 
90450 
90504 
90558 
.90612 
90666 

90 720 

90773 
90827 

.90 881 

90934 
90988 

91 041 
91094 

.91 148 
91 201 
91254 

91307 
91360 

•91 413 
91 466 

91519 
91572 
91 624 

.91 677 

91 730 
91 782 

91834 
91887 

•91 939 

91 991 
92044 
92096 

92 148 



•92 454 
92505 
92557 
92 609 
92 660 

.92711 
92763 
92814 
92865 
92.916 



•92459 
92511 
92562 
92614 
92665 

.92 716 
92 768 
92819 
92 870 
92921 



•90 347 
90401 

90455 
90509 

90563 
.90617 
90671 
90725 

90779 
90832 

.90 886 
90940 

90993 
91 046 
91 100 

•91 153 
91 206 
91259 
91 312 
91365 

.91 418 
91471 
91524 
91577 
91630 

.91 682 

91735 
91787 
91 840 

91 892 

.91944 
91997 
92049 

92 lOI 
92153 
.92 205 
92257 
92309 
92361 
92412 

.92464 
92 516 

92567 
92619 
92 670 

.92 722 .92 727 

92773 92778 

92 824 92 829 

92875 92881 

92927 92932 



•90 352 
90407 
90461 
90515 
90569 
•90 623 
90677 
90730 
90784 
90838 

.90 891 

90945 
90998 
91 052 
91 105 

.91 158 
91 212 
91 265 
91318 
91 371 

.91 424 

91477 
91529 
91582 
91635 
.91 687 
91740 

91793 
91845 
91897 



•90 358 
90412 
90466 
90520 
90574 
.90 628 
90682 
90736 
90789 
90843 

•90 897 
90950 
91004 
91057 
91 no 

.91 164 
91 217 
91 270 
91323 
91376 

.91 429 
91 482 
91535 
91587 
91 640 

.91 693 

91745 
91798 
91850 
91903 



.91 950 .91 955 
92 003 92007 
92054 92059 
92106 92 III 
92 158 92 163 

.92210 .92215 
92 262 92 267 
92314 92319 
92366 92371 
92418 92423 



.92 469 
92521 
92572 
92624 
92675 



.92 474 
92526 
92578 
92629 
92681 
.92 732 

92783 
92834 
92886 
92937 



LOGS. 5 PL. 
8. 



(26) 



FIVE PLACE LOGARITHMS. 



8. 
6 PL. LOGS. 



No. 

8.50 

•51 

•52 
•53 
•54 

8.55 
.56 

■57 
.58 

•59 

8.60 

.61 

.62 

•63 
.64 

8.65 
.66 
.67 
.68 
.69 

8.70 

•71 
.72 

■73 

•74 

8.75 

.76 

•77 
.78 

■79 

8.80 

.81 
.82 

•83 

.84 

8.85 
.86 
.87 



8.90 

.91 
.92 
•93 
■94 
8.95 
.96 

•97 
.98 

■99 



8 



.92 942 
92993 
93044 
93095 
93146 

■93 197 
93247 
93298 
93 349 
93 399 



•92 947 
92998 

93049 
93 100 

93 151 
■93 202 
93252 
93303 
93 354 
93404 



■93450.93455 
93500 93505 
93551 93556 
93601 93606 

93651 93656 



•93 702 
93752 
93802 
93852 
93902 

•93 952 
94 002 

94052 

94 101 

94 151 
.94 201 
94250 
94300 
94 349 
94 399 

■94 448 
94498 

94 547 
94596 

94645 
•94 694 
94 743 
94792 
94841 
94890 

■94 939 
94988 
95036 
95085 
95134 
.95 182 
95231 
95279 
95328 
95376 



■93 707 

93 757 
93807 

93857 
93907 

•93 957 
94007 

94057 

94 106 
94156 

.94 206 
94255 
94305 
94 354 
94404 

•94 453 
94503 
94552 
94601 
94650 

•94 699 
94748 

94 797 
94846 
94895 

•94 944 
94 993 
95041 
95090 

95139 
•95 187 
95236 
95284 
95332 
95381 



.92952 
93 003 

93054 
93105 
93156 

•93 207 
93258 
93308 
93 359 
93409 

•93 460 
93510 
93561 

93 611 
93661 

•93 712 
93762 
93812 
93862 
93912 

•93 962 
94012 
94062 

94 HI 
94 161 

.94 21 1 
94260 
94310 

94 359 
94409 

.94458 
94507 
94 557 
94606 

94655 

•94 704 

94 753 
94802 
94851 
94900 

•94 949 
94998 
95046 
95095 
95143 

•95 192 
95240 
95289 

95 337 
95386 



•92957 
93008 

93059 
93 HO 

93 161 

■93 212 
93263 
93313 
93364 
93414 

•93 465 
93515 
93566 
93616 
93666 

•93717 
93767 
93817 
94867 
93917 

•93 967 
94017 
94067 

94 1 16 
94 1 56 

.94216 
94265 
94315 
94364 
94414 

•94463 
94512 
94562 

94 61 1 
94660 

•94 709 
94758 
94807 
94856 
94905 

■94 954 

95 003 

95051 
95 loo 
95148 

•95 197 
95245 
95294 
95342 
95390 



.92 962 

93013 
93064 

93115 
93166 

•93217 
93268 
93318 

93369 
93420 

•93 470 
93520 

93571 
93621 

93671 

■93 722 
93772 
93822 
93872 
93922 

•93 972 
94022 
94072 

94 121 
94171 

.94 221 
94270 
94320 

94369 
94419 

.94 468 
94517 
94567 
94616 
94665 

•94 714 
94763 
94812 
94861 
94910 

•94 959 
95007 

95056 
95105 

95153 
.95 202 
95250 
95299 

95 347 
95 395 



.92 967 
93018 
93069 
93120 
93 171 
.93 222 
93273 
93323 
93 374 
93425 

•93 475 
93526 
93576 
93 626 
93676 

•93 727 

93 777 
93827 

93877 
93927 

•93 977 
94027 

94077 

94 126 
94176 

.94 226 
94275 
94325 

94 374 
94424 

•94473 
94522 

94571 
94621 
94670 

■94 719 
94768 
94817 
94866 
94915 

■94 963 
95012 
95061 

95 109 
95158 
■95 207 
95255 
95303 
95352 
95400 



■92 973 
93024 

93075 
93125 
93176 
■93 227 
93278 
93328 

93 379 
93430 

•93 480 
93531 
93581 
93631 
93682 

•93 732 
93782 
93832 
93882 
93932 

•93 982 
94032 
94082 

94131 

94 181 

•94 231 
94280 
94330 
94 379 
94429 

•94 478 
94527 
94576 
94626 

94675 

•94 724 

94 773 
94822 

94871 
94919 

.94 968 
95017 

95 066 
95114 
95163 
■95211 
95 260 
95308 
95 357 
95405 



.92 978 
93029 
93080 
93131 
93181 

•93 232 
93283 
93 334 
93384 
93 435 

■93 485 
93536 
93586 
93636 
93687 

■93 737 
93787 
93837 
93887 

93 937 

■93 987 
94037 
94086 
94136 

94 186 

•94 236 
94285 
94 335 
94384 

94 433 

•94 483 
94532 
94581 
94630 
94680 

•94 729 
94778 
94827 
94876 
94924 

•94 973 

95 022 

95071 
95 119 
95168 

.95 216 
95265 
95313 
95361 
95410 



.92983 .92988 

93034 93039 
93085 93090 
93 136 93 141 
93 186 93 192 



•93 237 
93288 

93 339 
93389 
93440 

•93 490 
93541 
93591 
93641 
93692 

•93 742 
93792 
93842 
93892 
93942 

■93 992 
94042 

94091 
94141 
94191 
.94 240 
94290 
94340 
94389 
94438 

.94 488 

94 537 
94586 

94635 
94685 

■94 734 
94783 
94832 
94880 
94929 

•94 978 
95027 

95075 

95 124 
95173 
.95 221 
95270 
95318 
95366 
95415 



(27) 



.93 242 
93293 
93 344 
93 394 
93 445 

•93 495 
93546 
93596 
93646 

93697 

•93 747 
93 797 
93847 
93897 

93 947 

•93 997 
94047 
94096 

94146 
94196 

•94 245 
94295 

94 345 
94 394 
94 443 

•94 493 
94542 
94591 
94640 
94689 

•94 738 
94 7.87 
94836 
94885 

94 934 

■94 983 
95032 

95 080 
95 129 
95177 
.95 226 
95274 
95323 
95371 
95419 

5 PL. 



INTP. 
TAB. 



LOGS. 
8. 



9. 
LOGS. 5 PL. 



FIVE PLACE LOGARITHMS. 



No. 


.01234 


5 67 8 9 


INTP. 
TAB. 

5 

I 
I 
2 
2 


. 9.00 

.OI 
.02 

•03 
.04 


•95 424 -95 429 -95 434 -95 439 '95 444 
95472 95 477 95482 95487 95492 
95521 95525 95530 95535 95540 

95569 95 574 95578 95583 95588 
95617 95622 95626 95631 95636 


•95 448 .95 453 •gs 458 .95 463 .95 468 

95 497 95501 95506 95511 95516 
95 545 95550 95 554 95 559 95564 
95 593 95598 95602 95607 95612 
95641 95646 95650 95655 95660 


9-05 
.06 
.07 
.08 
.09 


•95 66^ ^95 670 .95 674 ^95 679 ^95 684 
95713 95718 95722 95727 95732 
95 761 95 766 95 770 95 775 95 78° 
95 809 95 813 95 818 95 823 95 828 
95856 95861-95866 95871 95875 


.95 689 .95 694 .95 698 .95 703 .95 708 
95 737 95 742 95 746 95 751 95 756 
95785 95789 95 794 95 799 95804 
95832 95837 95842 95847 95852 
95880 95885 95890 95895 95899 


3 
3 
4 
4 
5 


9.10 

.11 

.12 

■13 
.14 


.95904 .95909 .95914 ^95918 .95923 
95952 95 957 95961 95966 95971 
95 999 96 004 96009 96014 96019 
96047 96052 96057 96061 96066 
96095 96099 96104 96109 96114 


•95 928 .95 933 .95 938 .95 942 .95 947 
95976 95980 95985 95990 95995 
96023 96028 96033 96038 96042 
96071 96076 96080 96085 96090 
96 118 96123 96128 96133 96137 


4 

I 
1 
2 


9.I5 
.16 

•17 
.18 
.19 


.96 142 .96 147 .96 152 .96 156 .96 161 
96190 96194 96199 96204 96209 
96237 96242 96246 96251 96256 
96 284 96 289 96 294 96 298 96 303 
96332 96336 96341 96346 96350 


.96166 .96171 .96175 .96180 .96185 
96 213 96 2i8 96 223 96 227 96 232 
96 261 96 265 96 270 96 275 96 280 
96308 96313 96317 96322 96327 
96355 96360 96365 96369 96374 


2 
2 
3 
3 
4- 


9.20 

.21 

.22 

■23 

.24 


•96 379 96 384 -96 388 .96 393 .96 398 
96426 96431 96435 96440 96445 
96473 96478 96483 96487 96492 
96520 96525 96530 96534 96539- 
96567 96572 96577 96581 96586 


.96402 .96407 .96412 .96417 .96421 
96450 96454 96459 96464 96468 
96497 96501 96506 96511 96515 
96544 96^48 96553 96558 96562, 
96 591 96 595 96 600 96 605 96 609 


5 
I 

1 
2 
2 


9.25 
.26 
.27 
.28 
.29 


.96 614 .96 619 .96 624 .96 628 .96 633 
96 661 96 666 96 670 96 675 96 680 
96708 96713 96717 96722 96727 

96755 96759 96764 96769 96774 
96802 96806 96 811 96816 96820 


.96638 .96642 .96647 .96652 .96656 
96 685 96 689 96 694 96 699 96 703 

96731 96736 96741 96745 96750 
96778 96783 96788 96792 96797 
96 825 96 830 96 834 96 839 96 844 


3 
3 
4 
4 
5 


9.30 

•3' 

•32 
•33 
•34 


.96848 .96853 .96858 .96862 .96867 
96895 96900 96904 96909 96914 
96942 96946 96951 96956 96960 
96988 96993 96997 97 002 97007 
97035 97039 97044 97049 97053 


.96872 .96876 .96881 .96886 .96890 
96918 96923 96928 96932 96937 

96965 96970 96974 96979 96984 
97 01 1 97016 97021 97025 97030 
97058 97063 97067 97072 97077 


4 

1 
1 
2 


9-35 
•36 
•37 
.38 
■39 


.97081 .97086 .97090 .97095 .97100 
97128 97132 97137 97142 97146 
97174 97179 97183 97188 97192 
97220 97225 97230 97234 97239 
97 267 97 271 97 276 97 280 97 285 


.97 104 .97 109 .97 114 .97 118 .97 123 
97 151 97155 97160 97165 97169 
97197 97202 97206 97211 97216 
97243 97248 97253 97257 9/262 
97290 97294 97299 97304 97308 


2 
2 
3 
3 
4 


9.40 

.41 

.42 

■43 
.44 


■97313 .97317 ^97322 .97327 .97331 
97 359 97364 97368 97 373 97 377 
97405 97410 97414 97419 97424 
97451 97456 97460 97465 97470 
97497 97502 97506 97511 97516 


•97 336 -97 340 .97 345 ^97 35o .97 354 
97382 97387 97391 97.396 97400 
97428 97433 97437 97442 97 447 
97 474 97 479 97 483 97 488 97 493 
97520 97525 97529 97534 97539 


6 

I 
1 
2 
2 


9-45 
.46 

•47 
.48 
■49 


•97 543 ^97 548 .97 552 .97 557 ^97 562 
97589 97 594 97598 97603 97607 
97635 97-640 97644 97649 97653 
97681 97685 97690 97695 97699 
97727 97731 97736 97740 97745 


•97566 .97571 ^97 575 -97580 .97585 
97612 97617 97621 97626 97630 
97 658 97 663 97 667 97 672 97 676 
97704 97708 97713 97717 97722 
97 749 97 754 97 759 97 763 97 768 


3 
3 
4 
4 

5 



LOGS. 6 PL. 
9. 



(28) 



FIVE PLACE LOGARITHMS. 



5 PL. LOGS. 



No. 


01234 


56789 


INTP, 
TAB. 


9.50 

•51 

•52 
■53 
•54 


•97 772 -97 777 ^97 782 .97 786 .97 79i 
97818 97823 97827 97832 97836 
97864 97868 97873 97877 97882 
97909 97914 97918 97923 97928 
97 955 97 959 97 9^4 97 968 97 973 


•97 795 -97800 .97804 ,97809 .97813 
97841 97845 97850 97855 97859 
97 886 97 891 97 896 97 900 97 905 

97932 97 937 97 941 97 946 97 950 
97978 97982 97987 97991 97996 


4 

1 
I 
2 


9-55 
.56 

•57 
•58 
•59 


.98 000 .98 005 .98 009 .98 014 .98 019 
98046 98050 98055 98059 98064 
98091 98096 98100 98105 98109 
98 137 98 141 98 146 98 150 98 155 
9_8 182 98186 98191 98195 98200 


.98 023 .98 028 .98 032 .98 037 .98 041 
98068 98073 98078 98082 98087 
98114 98118 98123 98127 98132 
98 159 98 164 98 168 98 173 98 177 
98 204 98 209 98 214 98 218 98 223 


2 
2 
3 

3 
4 


9.60 

.61 
.62 
■63 
.64 


.98 227 .98 232 .98 236 .98 241 .98 245 
98 272 98 277 98 281 98 286 98 290 
98318 98322 98327 98331 98336 

98363 98367 98372 98376 98381 
98408 98412 98417 98421 98426 


.98250 .98254 .98259 .98263 .98268 
98295 98299 98304 98308 98313 
98340 98345 98349 98354 98358 

98385 98390 98394 98399 98403 
98430 98435 98439 98444 98448 


5 
1 

1 
2 
2 


9-65 

.66 
.67 
.68 
.69 


.98453 .98457 .98462 .98466 .98471 
98498 98502 98507 98511 98516 

98543 98547 98552 98556 98561 
98588 98592 98597 98601 98605 
98 632 98 637 98 641 98 646 98 650 


.98475 .98480 .98484 .98489 .98493 
98520 98525 98529 98534 98538 

98565 98570 98574 98579 98583 
98610 98614 98619 98623 98628 
98655 98659 98664 98668 98673 


3 
3 

4 
4 
5 


9.70 

■71 
.72 

•73 
•74 


.98677 .98682 .98686 .98691 .98695 
98722 98726 98731 98735 98740 
98767 98771 98776 98 7S0 98784 
98 811 98816 98820 98825 98829 
98856 98860 98865 98869 98874 


.98700 .98704 .98709 .98713 .98717 

98744 98749 98753 98758 98762 
98789 98793 98798 98802 98807 
98834 98838 98843 98847 98851 
98878 98883 98887 98892 98896 


4 


I 

1 
2 


9-75 
.76 

•77 
.78 

•79 


.98900 .98905 .98909 .98914 ,98918 
98945 98949 98954 98958 98963 
98989 98994 98998 99 003 99007 
99034 99038 99043 99047 99052 
99078 99083 99087 99092 99096 


.98923 .98927 .98932 .98936 .98941 
98967 98972 98976 98981 98985 
99012 99016 99021 99025 99029 
99056 99061 99065 99069 99074 
99100 99105 99109 99 114 99118 


2 
2 
3 
3 
4 


9.80 

.81 
.82 

•83 
.84 


•99 123 .99 127 .99 131 .99 136 .99 140 
99 167 99 171 99 176 99 180 99 185 

99 2H 99216 99220 99224 99229 
99255 99260 99264 99269 99273 
99300 99304 99308 99313 99317 


•99 145 ^99 149 -99 154 -99 158 -99 162 - 
99 189 99 193 99 198 99 202 99 207 
99233 99238 99242 99247 99251 
99 277 99 282 99 286 99 291 99 295 
99322 99326 99330 99335 99339 


5 

1 
I 
2 
2 


9-85 
.86 

•87 
.88 
.89 


•99 344 .99348 ^99352 ^99 357 -99 361 
99388 99392 99396 99401 99405 
99432 99436 99441 99445 99449 
99476 99480 99484 99489 99493 
99520 99524 99528 99533 99537 


•99 366 .99 370 .99 374 ^99 379 -99 383 
99410 99414 99419 99423 99427 

99 454 99458 99463 99467 99471 
99498 99502 99506 99511 99515 

99542 99546 99550 99 555 99 559 


3 
3 
4 
4 
5 


9.90 

•91 
.92 

•93 
•94 


•99564 .99568 .99572 .99 577 ^99581 
99607 99612 99616 99621 99625 
99651 99656 99660 99664 99669 
99695 99699 99704 99708 99712 
99 739 99 743 99 747 99 752 99 756 


•99 585 ^99 590 .99 594 .99 599 ^99 603 
99 629 99 634 99 638 99 642 99 647 
99673 99677 99682 99686 99691 
99717 99721 99726 99730 99734 
99760 99765 99769 99 774 99778 


4 


1 
1 
2 


9-95 
.96 

•97 
.98 

•99 


■99 782 .99 787 ^99 791 .99 795 .99 800 
99826 99830 99835 99839 99843 
99 870 99 874 99 878 99 883 99 887 
99913 99917 99922 99926 99930 
99 957 99961 99965 99970 99974 


.99804 .99808 .99813 .99817 .99822 
99 848 99 852 99 856 99 861 99 865 
99891 99896 99900 99904 99909 
99 935 99 939 99 944 99 948 99952 
99978 99983 99987 99 991 99996 


2 
2 
3 
3 

4 



(29) 



5 PL. LOGS. 
9. 



1.-10. 
SQ. RTS. 



86 SQRS. 



SQUARE ROOTS AND SQUARES. 

Note. The table gives roots directly, squares by inverse interpolation. 



No. 



Interpola. 
for Thousandths. 



1.0 

.1 

.2 

•3 

•4 

i-S 
.6 

•7 
.8 

•9 

2.0 

.1 

.2 

•3 
■4 

2.5 

.6 



•9 

3.0 

.1 

.2 

•3 

•4 

3-5 
.6 

•7 
.8 

•9 

4.0 

.1 

.2 

•3 

•4 

4-5 
.6 

■7 



I.OOO 

1.049 
1.095 
1. 140 
1. 183 
1.225 
1.265 
1.304 
1.342 
1-378 

1.414 
1.449 
1.483 
1.517 
1.549 

1.581 
1.612 
1.643 
1-673 
1703 

1.732 
1. 761 
1.789 
1. 81 7 

1.844 

1.871 
1.897 
1.924 
1.949 
1-975 

2.000 

2.025 
2.049 
2.074 
2.098 

2.121 

2.145 
2.168 
2. 191 
2.214 



.005 
.054 
.100 

.145 
.187 

.229 
.269 
.308 

•345 
.382 

.418 

-453 
.487 
.520 
-552 

.584 
.616 
.646 
.676 
.706 

■735 
.764 

•792 
.819 

.847 

-873 
.900 

.926 
.952 
-977 



2.027 
2.052 
2.076 
2.100 

2.124 
2.147 
2.170 
2.193 
2.216 



.010 
-058 
.105 
.149 
.192 

-233 
•273 
■3" 
-349 
-386 

.421 
.456 
.490 
-523 
■556 
.587 
.619 
.649 
.679 
.709 

-738 
.766 

■794 
.822 
.849 

.876 

-903 
.929 

•954 
.980 

2.005 
2.030 
2.054 
2.078 
2.102 

2.126 
2.149 
2-173 
2^i95 
2.218 



1.015 
1.063 
1. 109 

1^153 
1. 196 

1^237 
1.277 

'•315 
'•353 
i^389 

1.425 
'•459 
'•493 
1.526 

'■559 
1.591 
1.622 
1.652 
1.682 
1. 712 

1-741 
1.769 
1.797 
1.825 
1.852 

1.879 
1.905 

1-931 

1-957 
1.982 

2.007 
2.032 
2.057 
2.081 
2.105 

2.128 
2.152 

2-175 
2.198 
2.220 



.020 
.068 
.114 
.158 
.200 

.241 
.281 
-319 
-356 
•393 

.428 
•463 
■497 
-530 
.562 

•594 
.625 
.655 
.685 
•715 

•744 
-772 
.800 

.828 
-855 
.881 
.908 

-934 
.960 

•985 

2.010 
2.035 
2.059 
2.083 
2.107 

2.131 

2.154 
2.177 
2.200 
2.223 



1.025 
1.072 
1.118 
1.162 
1.204 

1.245 
1.285 

1-323 
1.360 
1.396 

1.432 
1.466 
1.500 

1-533 
1.565 

1-597 
1.628 
1.658 
1.688 
1.718 

1.746 

1-775 
1.803 
1.830 
1-857 
1.884 
1.910 
1-936 
1.962 

1-987 

2.012 
2.037 
2.062 
2.086 
2.110 

2-133 
2.156 
2.179 
2.202 
2.225 



1 .030 ] 


-034 


-039 1 


1.077 


.082 


.086 1 


1.122 1 


.127 ] 


-131 1 


1.166 


.170 


-175 I 


1.208 ] 


.212 


.217 1 


1.249 


-253 


•257 1 


1.288 


.292 


.296 1 


1-327 


-330 


-334 1 


1.364 


-367 


-371 1 


1.400 


.404 


.407 I 


1-435 1 


-439 


.442 I 


1.470 


-473 


.476 1 


i-5°3 


.507 


.510 1 


1-536 


-539 


•543 1 


1.568 


'•572 


•575 1 


1.600 


-603 


.606 1 


1.631 


•634 


-637 1 


1.661 


.664 


.667 1 


1.691 


.694 


-697 1 


1.720 1 


-723 


.726 1 


1-749 


-752 


•755 1 


1.778 


.780 


•783 I 


1.806 


[.808 


[.8n 1 


1-8.33 


-836 


.838 I 


1.860 


.863 


.865 I 


1.887 ' 


.889 1 


.892 I 


1-913 


.916 


.918 I 


1-939 


.942 


■944 I 


1.965 1 


.967 ] 


.970 1 


1.990 


.992 


-995 1 



2.015 
2.040 
2.064 

2.088 

2.112 

2.135 
2.159 
2.182 
2.205 
2.227 



2.017 
2.042 
2.066 
2.090 
2.114 

2.138 
2.161 
2.184 
2.207 
2.229 



2.020 
2.045 
2.069 
2.093 
2.117 

2.140 
2.163 
2.186 
2.209 
2.232 



.044 
.091 
.136 
-179 
.221 

.261 

.300 

-338 

-375 
.411 

.446 
.480 

-513 
-546 
-578 
.609 
.640 
.670 
.700 
.729 

•758 
.786 
.814 
.841 
.868 

•895 
.921 

•947 
•972 
-997 

2.022 
2.047 
2.071 
2.095 
2.119 

2.142 
2.166 
2.189 
2.211 
2.234 



SQ. RTS. & SQRS. 
1.-10 



(30) 



SQUARE ROOTS AND SQUARES. i.-io. 

SQ. RTS. & SQRS. 



No. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Interpola. for 1 
Thousandths. 1 


5.0 


2.236 


2.238 


2.241 


2.243 


2.245 


.2.247 


2.249 


2.252 


2-254 


2.256 


3 


2 


.1 


2.258 


2.261 


2.263 


2.265 


2.267 


2.269 


2.272 


2.274 


2.276 


2.278 








.2 


2.280 


2.283 


2.285 


2.287 


2.289 


2.291 


2.293 


2.296 


2.298 


2300 


I 





•3 


2.302 


2.304 


2.307 


2.309 


2.311 


2.313 


2.315 


2.317 


2-319 


2.322 


1 




•4 


2.324 


2.326 


2.328 


2.330 


2.332 


2.335 


2.337 


2.339 


2.341 


2.343 


1 




S-S 


2-345 


2.347 


2-349 


2.352 


2.354 


2.356 


2.358 


2.360 


2.362 


2.364 


2 




.6 


2.366 


2.369 


2-371 


2-373 


2-375 


2.377 


2.379 


2.381 


2.383 


2-385 


2 




•7 


2-387 


2.390 


2.392 


2-394 


2.396 


2.398 


2.400 


2.402 


2.404 


2.406 


2 




.8 


2.408 


2.410 


2.412 


2.415 


2.417 


2.419 


2.421 


2.423 


2.425 


2.427 


2 


2 


•9 


2.429 


2-431 


2.433 


2.435 


2.437 


2.439 


2.441 


2.443 


2.445 


2.447 


3 


2 


6.0 


2.449 


2.452 


2.454 


2.456 


2.458 


2.460 


2.462 


2.464 


2.466 


2.468 


3 


2 


.1 


2.470 


2.472 


2.474 


2.476 


2.478 


2.480 


2.482 


2.484 


2.486 


2.488 








.2 


2.490 


2.492 


2.494 


2.496 


2.498 


2.500 


2.502 


2.504 


2.506 


2.508 


1 





•3 


2.510 


2.512 


2.514 


2.516 


2.518 


2.520 


2.522 


2.524 


2.526 


2.528 


1 




•4 


2.530 


2-532 


2-534 


2.536 


2.538 


2-540 


2.542 


2.544 


2.546 


2.548 


I 




6.5 


2.550 


2.551 


2-553 


2-555 


2-557 


2-559 


2.561 


2-563 


2.565 


2.567 


2 




.6 


2.569 


2-571 


2-573 


2-575 


2-577 


2-579 


2.581 


2.583 


2.585 


2.587 


2 




•7 


2.588 


2.590 


2.592 


2.594 


2.596 


2.598 


2.600 


2.602 


2.604 


2.606 


2 




.8 


2.608 


2.610 


2.612 


2.613 


2.615 


2.617 


2.619 


2.621 


2.623 


2.625 


2 


2 


•9 


2.627 


2.629 


2.631 


2.632 


2.634 


2.636 


2.638 


2.640 


2.642 


2.644 


3 


2 


7.0 


2.646 


2.648 


2.650 


2.651 


2-653 


2.655 


2.657 


2.659 


2.661 


2.663 


2 


1 


.1 


2.665 


2.666 


2.668 


2.670 


2.672 


2.674 


2.676 


2.678 


2.680 


2.681 








.2 


2.683 


2.685 


2.687 


2.689 


2.691 


2.693 


2.694 


2.696 


2.698 


2.700 








■3 


2.702 


2.704 


2.706 


2.707 


2.709 


2.711 


2.713 


2.71S 


2.717 


2.718 







■4 


2.720 


2.722 


2.724 


2.726 


2.728 


2.729 


2.731 


2-733 


2.735 


2.737 







7-5 


2.739 


2.740 


2.742 


2.744 


2.746 


2.748 


2.750 


2-751 


2.753 


2.755 






.6 


2.757 


2.759 


2.760 


2.762 


2.764 


2.766 


2.768 


2.769 


2.771 


2.773 






•7 


2-775 


2.777 


2.778 


2.780 


2.782 


2.784 


2.786 


2-787 


2.789 


2.791 , 






.8 


2-793 


2.795 


2.796 


2.798 


2.800 


2.802 


2.804 


2.805 


"2.807 


2.809 


2 




•9 


2.81 1 


2.8l2 


2.814 


2.816 


2.818 


2.820 


2.821 


2.823 


2.825 


2.827 


2 




8.0 


2.828 


2.830 


2.832 


2.834 


2.835 


2837 


2.839 


2.841 


2.843 


2.844 


2 


1 ■ 


.1 


2.846 


2.848 


2.850 


2.851 


2.853 


2.855 


2.857 


2.858 


2.860 


2.862 








.2 


2.864 


2.865 


2.867 


2.869 


2.871 


2.872 


2.874 


2.876 


2-877 


2.879 








•3 


2.881 


2.883 


2.884 


2.886 


2.888 


2.890 


2.891 


2.893 


2.895 


2.897 







■4 


2.898 


2.900 


2.902 


2.903 


2.905 


2.907 


2.909 


2.910 


2.912 


2-914 







8.S 


2.915 


2.917 


2.919 


2.921 


2.922 


2.924 


2.926 


2.927 


2.929 


2.931 






.6 


2-933 


2.934 


2.936 


2.938 


2-939 


2.941 


2.943 


2.944 


2.946 


2.948 






•7 


2.950 


2.951 


2.953 


2-955 


2.956 


2.958 


2.960 


2.961 


2.963 


2.965 






.8 


2.966 


2.968 


2.970 


2.972 


2.973 


2-975 


2.977 


2.978 


2.980 


2.982 


2 




•9 


2.983 


2.985 


2.987 


2.988 


2.990 


2.992 


2.993 


2.995 


2.997 


2.998 


2 




9.0 


3.000 


3.002 


3-003 


3.005 


3.007 


3.008 


3.010 


3.012 


3-013 


3015 


2 


1 


.1 


3-017 


3-Oi8 


3.020 


3.022 


3.023 


3.025 


3.027 


3.028 


3030 


3.032 








.2 


3-033 


3-035 


3-036 


3-038 


3.040 


3-041 


3.043 


3-045 


3-046 


3.048 








■3 


3.050 


3-051 


3-053 


3-055 


3-056 


3.058 


3.059 


3.061 


3-063 


3-064 







•4 


3.066 


3.068 


3-069 


3-071 


3.072 


3-074 


3.076 


3-077 


3-079 


3.081 







9-S 


3.082 


3.084 


3.085 


3.087 


3.089 


3.090 


3.092 


3-094 


3.095 


3097 






.6 


3.098 


3.100 


3.102 


3-103 


3.105 


3.106 


3.108 


3.110 


3.1 11 


3-113 






•7 


3- "4 


3.116 


3.118 


3-119 


3.121 


3.122 


3.124 


3.126 


3.127 


3.129 






.8 


3-130 


3-132 


3-134 


3-135 


3.137 


3-138 


3.140 


3.142 


3-143 


3-145 


2 




•9 


3.146 


3.148 


3-150 


3-151 


3.153 


3.154 


3.156 


3-158 


3-159 


3.161 


2 





(31) 



SQ. RTS. 



86 SQRS. 
1.-10. 



lO.-lOO. 
SQ. RTS. 



& SQRS. 



SQUARE ROOTS AND SQUARES. 



No. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Interpola. 1 
for Hundredths. | 


10. 


3.162 


3-178 


3-194 


3.209 


3-225 


3.240 


3.256 


3-271 


3.286 


3-302 


16 


14 


II. 


3-317 


3-332 


3-347 


3-362 


3-376 


3-391 


3-406 


3-421 


3-435 


3450 


2 


I 


12. 


3464 


3-479 


3-493 


3.507 


3-521 


3-536 


3-550 


3564 


3-578 


3-592 


3 


3 


13- 


3.606 


3-619 


3-633 


3-647 


3-661 


3-674 


3.688 


3-701 


3-715 


3-728 


5 


4 


14. 


3-742 


3-755 


3.768 


3.782 


3-795 


3.808 


3.821 


3-834 


3-847 


3.860 


6 


6 


15- 


3-873 


3.886 


3-899 


3.912 


3-924 


3-937 


3-950 


3.962 


3-975 


3-987 


8 


7 


16. 


4.000 


4.012 


4.025 


4-037 


4.050 


4.062 


4-074 


4.087 


4-099 


4.1 1 1 


10 


8 


17- 


4-123 


4-135 


4.147 


4-159 


4.171 


4.183 


4-195 


4.207 


4-219 


4-231 


II 


10 


18. 


4-243 


4.254 


4.266 


4.278 


4.290 


4.301 


4-313 


4-324 


4-336 


4-347 


13 


II 


19. 


4-359 


4-370 


4.382 


4-393 


4.405 


4.416 


4-427 


4-438 


4-450 


4.461 


14 


13 


20. 


4-472 


4483 


4-494 


4.506 


4-517 


4.528 


4-539 


4-550 


4.561 


4-572 


12 


10 


21. 


4-583 


4-593 


4.604 


4.615 


4.626 


4-637 


4.648 


4.658 


4.669 


4.680 


I 


I 


22. 


4.690 


4.701 


4.712 


4.722 


4-733 


4-743 


4-754 


4-764 


4-775 


4-785 


2 


2 


23. 


4.796 


4.806 


4.817 


4.827 


4-837 


4.848 


4.858 


4.868 


4.879 


4.889 


4 


3 


24. 


4-899 


4-909 


4.919 


4-930 


4.940 


4.950 


4.960 


4.970 


4.980 


4.990 


5 


4 


25. 


5.000 


5.010 


5.020 


5.030 


5-040 


5.050 


5.060 


5.070 


S-079 


5.089 


6 


5 ■ 


26. 


S-099 


5.109 


5. 119 


5.128 


5-138 


5.148 


5-158 


5.167 


5-177 


5.187 


7 


6 


27. 


5.196 


5.206 


5.215 


5-225 


5-235 


5-244 


5-254 


5-263 


5-273 


5.282 


. 8 


7 


28. 


5.292 


S-301 


5-310 


5-320 


5-329 


5-339 


5-348 


5-357 


5-367 


5-376 


10 


8 


29. 


5-385 


5-394 


5.404 


5-413 


5-422 


5-431 


5-441 


5.450 


5-459 


5.468 


II 


9 


30. 


5-477 


5.486 


5-495 


5.505 


5-514 


5-523 


5-532 


S-541 


5-550 


5-559 


9 


8 


31- 


5.568 


5-577 


5-586 


5-595 


5,604 


5.612 


5.621 


5.630 


S-639 


5.648 


I 


I 


32. 


5-657 


5.666 


5-675 


5.683 


5.692 


5-701 


5.710 


5.718 


5-727 


5-736 


2 


2 


33- 


5-745 


5-753 


5-762 


5-771 


5-779 


5-788 


5-797 


5.805 


5.814 


5.822 


3 


2 


34- 


5-831 


5-840 


5-848 


5-857 


5.865 


5-874 


5.882 


5.891 


5-899 


5.908 


.4 


3 


35- 


5-916 


5-925 


5-933 


5-941 


5-950 


5-958 


5-967 


5-975 


5-983 


5.992 


5 


4 


36. 


6.000 


6.008 


6.017 


6.025 


6-033 


6.042 


6.050 


6.058 


6.066 


6-075 


5 


5 


37- 


6.083 


6.091 


6.099 


6.107 


6.1:6 


6.124 


6.132 


6.140 


6.148 


6.156 


6 


6 


38- 


6.164 


6-173 


6.181 


6.189 


6.197 


6.205 


6.213 


6.221 


6.229 


6.237 


7 


6 


39- 


6.245 


6.253 


6.261 


6.269 


6.277 


6.285 


6.293 


6.301 


6.309 


6.317 


8 


7 


40. 


6-325 


6-332 


6.340 


6.348 


6.356 


6.364 


6.372 


6.380 


6.387 


6-395 


8 


7 


41. 


6.403 


6.41 1 


6.419 


6.427 


6.434 


6.442 


6.450 


6.458 


6.465 


6.473 


I 


I 


42. 


6.481 


6.488 


6.496 


6.504 


6.512 


6.519 


6.527 


6-535 


6.542 


6.550 


2 


I 


43- 


6-557 


6.56s 


6-573 


6.580 


6.588 


6.595 


6.603 


6.61 1 


6.618 


6.626 


2 


2 


44. 


6.633 


6.641 


6.648 


6.656 


6.663 


6.671 


6.678 


6.686 


6.693 


6.701 


3 


3 


45- 


6.708 


6.716 


6.723 


6.731 


6.738 


6.745 


6.753 


6.760 


6.768 


6-775 


4 


4 


46. 


6.782 


6.790 


6-797 


6.804 


6.812 


6.819 


6.S26 


6.834 


6.841 


6.848 


5 


4 


47- 


6.856 


6.863 


6.870 


6.877 


6.885 


6.892 


6.899 


6.907 


6.914 


6.921 


6 


5 


48. 


6.928 


6-935 


6.943 


6.950 


6.957 


6.964 


6.971 


6-979 


6.986 


6.993 


6 


6 


49. 


7.000 


7.007 


7.014 


7.021 


7.029 


7.036 


7-043 


7.050 


7-057 


7.064 


7 


6 


"rt 


S. & SQRS. 






(32) 















lO.-lOO. 



SQUARE ROOTS AND 



SQUARES. 10.-100. 

SQ. RTS. & SQRS. 



No. 





I 2 


3 


4 


5 


6 


7 


8 


9 


Interpola. for 1 
Hundredttis. 1 


50. 


7.071 


7.078 7.085 


7.092 


7.099 


7.106 


7-"3 


7.120 


7.127 


7-"34 


7 


6 


51- 


7-141 


7-148 7-155 


7.162 


7.169 


7.176 


7-183 


7.190 


7-197 


7.204 


I 


1 


52. 


7.2II 


7.218 7.225 


7.232 


7-239 


7-246 


7-253 


7.259 


7.266 


7-273 


1 


1 


53- 


7.280 


7.287 7.294 


7-301 


7-308 


7-314 


7-321 


7-328 


7-335 


7-342 


2 


2 


54- 


7-348 


7-355 7-362 


7-369 


7-376 


7.382 


7-389 


7-396 


7403 


7.409 


3 


2 


55- 


7.416 


7.423 7.430 


7-436 


7-443 


7-450 


7-457 


7-463 


7.470 


7-477 


4 


3 


56. 


7-483 


7.490 7.497 


7.503 


7.510 


7-517 


7-523 


7-530 


7-537 


7-543 


4 


4 


57- 


7-55° 


7.556 7.563 


7.570 


7.576 


7-583 


7-589 


7-596 


7-603 


7.609 


5 


4 


58- 


7.616 


7.622 7.629 


7-635 


7.642 


7.649 


7-655 


7.662 


7.668 


7.675 


6 


5 


59- 


7.681 


7.688 7.694 


7.701 


7.707 


7-7H 


7.720 


7.727 


7-733 


7.740 


6 


5 


60. 


7.746 


7-752 7.759 


7-765 


7-772 


7.778 


7-785 


7.791 


7-797 


7.804 


7 


6 


61. 


7.810 


7.817 . 7.823 


7.829 


7.836 


7.842 


7.849 


7-855 


7.861 


7.868 


I 


1 


62. 


7.874 


7.880 7.887 


7-893 


7.899 


7.906 


7.912 


7.918 


7-925 


7-931 


1 


1 


63- 


7-937 


7-944 7-950 


7-956 


7.962 


7-969 


7-975 


7.981 


7.987 


7-994 


2 


2 


64. 


8.000 


8.006 8.012 


8.019 


8.025 


8.031 


8.037 


8.044 


8.050 


8.056 


3 


2 


65. 


8.062 


8.068 8.07s 


8.081 


8.087 


8.093 


8.099 


8.106 


8.112 


8.118 


4 


3 


66. 


8.124 


8.130 8.136 


8.142 


8.149 


8-155 


8.161 


8.167 


8-173 


8.179 


4 


4 


67. 


8.185 


8.191 8.198 


8.204 


8.210 


8.216 


8.222 


8.228 


8.234 


8.240 


5 


4 


68. 


8.246 


8.252 8.258 


8.264 


8.270 


8.276 


8.283 


8.289 


8.295 


8.301 


6 


5 


69. 


8.307 


8.313 8.319 


8.325 


8-331. 


8.337 


8-343 


8-349 


8-355 


8-361 


6 


S 


70. 


8.367 


8-373 8.379 


8-385 


8.390 


8-396 


8.402 


8.408 


8.414 


8.420 


6 


5 


71- 


8.426 


8.432 8.438 


8.444 


8.450 


8.456 


8.462 


8.468 


8-473 


8.479 


1 


I 


72- 


8.485 


8.491 8.497 


8.803 


8.509 


8-515 


8.521 


8.526 


8-532 


8.538 


1 


I 


73- 


8.544 


8.550 8.556 


8.562 


8.567 


8-573 


8-579 


8.585 


8.591 


8-597 


2 


2 


74- 


8.602 


8.608 8.614 


8.620 


8.626 


8.631 


8-637 


8.643 


8.649 


8.654 


2 


2 


75- 


8.660 


8.666 8.672 


8.678 


8.683 


8.689 


8-695 


8.701 


8.706 


8.712 


3 


3 


76. 


8.718 


8.724 8.729 


8-735 


8.741 


8.746 


8-752 


8.758 


8.764 


8.769 


4 


3 


77- 


8-77S 


8.781 8.786 


8.792 


8.798 


8.803 


8.809 


8.815 


8.820 


8.826 


4 


4 


78. 


8.832 


8.837 8.843 


8.849 


8.854 


8.860 


8.866 


8.871 


8.877 


8.883 


5 


4 


79- 


8.888 


8.894 8.899 


8.905 


8.911 


8.916 


8.922 


8.927 


8-933 


8-939 


5 


5 


80. 


8.944 


8.950 8.955 


8.961 


8.967 


8.972 


8-978 


8.983 


8.989 


8.994 


6 


5 


8i. 


9.000 


9.006 9.01 1 


9.017 


9.022 


9.028 


9-033 


9-039 


9.044 


9.050 


1 


1 


82. 


9-055 


9.061 9.066 


9.072 


9-077 


9.083 


9.088 


9.094 


9.099 


9.105 


1 


1 


83. 


9.110 


9.116 9.121 


9.127 


9.132 


9-138 


9-143 


9.149 


9.154 


9.160 


2 


2 


84. 


9- 1 65 


9.171 9.176 


9.182 


9.187 


9.192 


9.198 


9.203 


9.209 


9-214 


2 


2 


85. 


9.220 


9.225 9.230 


9.236 


9.241 


9.247 


9.252 


9-257 


9.263 


9.268 


3 


3 


86. 


9.274 


9.279 9.284 


9.290 


9-295 


9-301 


9.306 


9-3II 


9-317 


9.322 


4 


3 


87. 


9-327 


9-333 9-338 


9-343 


9-349 


9-354 


9-359 


9-365 


9-370 


9-375 


4 


4 


88. 


9-381 


9.386 9.391 


9-397 


9402 


9-407 


9-413 


9.418 


9-423 


9-429 


5 


4 


89. 


9-434 


9-439 9-445 


9.450 


9-455 


9.460 


9.466 


9-471 


9-476 


9.482 


5 


5 


90. 


9.487 


9.492 9.497 


9.503 


9.508 


9-513 


9-518 


9-524 


9-529 


9-534 


5 




91- 


9-539 


9-545 9-550 


9-555 


9.560 


9-566 


9-571 


9-576 


9.581 


9.586 


I 




92. 


9.592 


9.597 9.602 


9.607 


9.612 


9.618 


9-623 


9.628 


9-633 


9-638 


I 




93- 


9.644 


9.649 9.654 


9-659 


9.664 


9.670 


9-675 


9.680 


9.685 


9.690 


2 




94- 


9-695 


9.701 9.706 


9-7" 


9.716 


9.721 


9.726 


9-731 


9-737 


9.742 


2 




95- 


9-747 


9.752 9.757 


9.762 


9.767 


9.772 


9.778 


9-783 


9-788 


9-793 


3 




96. 


9-798 


9.803 9.808 


9-813 


9.818 


9-823 


9.829 


9-834 


9-839 


9.844 


3 




97- 


9.849 


9.854 9.859 


9.864 


9.869 


9.874 


9-879 


9.884 


9.889 


9-894 


4 




98. 


9.899 


9-905 9-9IO 


9-915 


9.920 


9-925 


9-930 


9-935 


9.940 


9-945 


4 




99- 


9.950 


9.955 9.960 


9-965 


9.970 


9-975 


9.980 


9.985 


9.990 


9-995 


5 





(33) 



SQ. RTS. & SQRS. 
10.-100. 



RECIPROCALS. 



RECIP. 



No. 


01234 


56789 


INTERPOLATION 
TABLES. 


1.00 

.01 
.02 

•03 
.04 


0.9990 0.9980 0.9970 0.9960 

0.9901 9891 9881 9872 9862 

9804 9794 9785 9775 9766 

9709 9699 9690 9680 9671 

9615 9606 9597 9588 9579 


0.9950 0.9940 0.9930 0.9921 0.991 1 
9852 9843 9833 9823 9814 

9756 9747 9737 9728 9718 
9662 9653 9643 9634 9625 
9569 9560 9551 9542 9533 


-85- 

9 
17 
26 

34 


75-65-65 

876 
15 13 II 
23 20 17 
30 26 22 


1.05 
.06 
.07 
.08 
.09 


0.9524 0.95 1 5 0.9506 0.9497 0.9488 
9434 9425 9416 9407 9399 
9346 9337 9328 9320 93" 
9259 9251 9242 9234 9225 
9174 9166 9158 9149 9141 


0.9479 0.9470 0.9461 0.9452 0.9443 
9390 9381 9372 9363 9355 
9302 9294 9285 9276 9268 
9217 9208 9200 9191 9183 
9132 9124 9116 9107 9099 


43 38 33 28 
51 45 39 33 
60 53 46 39 
68 60 52 44 
77 68 59 50 

-45-35-25-22 

5432 

9 7 5 4 
14 n 8 7 
18 14 10 9 


1.0 
.1 

.2 
•3 
•4 


0.9901 0.9804 0.9709 0.9615 

0.9091 9009 8929 8850 8772 

8333 8264 8197 8130 8065 

7692 7634 7576 7519 7463 

7143 7092 7042 6993 6944 


0.9524 0.9434 0.9346 0.9259 0.9174 
8696 8621 8547 8475 8403 
8000 -7937 7874 7813 7752 

7407 7353 7299 7246 7194 
6897 6849 6803 6757 67 I I 


I -5 
.6 

•7 
.8 

•9 


0.6667 0.6623 0-6579 0.6536 0.6494 
6250 621 1 6173 6135 6098 
5882 5848 5814 5780 5747 

5556 5525 5495 5464 5435 
5263 5236 5208 5181 5155 


0.6452 0.6410 0.6369 0.6329 0.6289 
6061 6024 5988 5952 5917 
5714 5682 5650 5618 5587 
5405 5376 5348 5319 5291 
5128 5102 5076 5051 5025 


23 
27 
32 
36 
41 


18 13 II 
21 15 13 
25 18 15 
28 20 18 
32 23 20 


2.0 

.1 

.2 
•3 
4 


0.5000 0.4975 0.4950 0.4926 0.4902 
4762 4739 4717 4695 4673 

4545 4525 4505 4484 4464 
4348 4329 4310 4292 4274 
4167 4149 4132 4115 4098 


0.4878 0.4854 0.4831 0.4808 0.4785 
4651 4630 4608 4587 4566 
4444 4425 4405 4386 4367 
4255 4237 4219 4202 4184 
4082 4065 4049 4032 4016 


-18-16-14-12 
2211 
4332 

5 5 4 4 
7665 


2-5 

.6 

•7 
.8 

•9 


0.4000 0.3984 0.3968 0.3953 0.3937 
3846 3831 3817 3802 3788 
3704 3690 3676 3663 3650 
357« 3559 3545 3534 3521 
3448 3436 3425 3413 3401 


0.3922 0.3906 0.3891 0.3876 0.3861 

3774 3759 3745 373i 371? 
3636 3623 3610 3597 3584 

3509 3497 3484 3472 3460 
3390 3378 3367 3356 3344 


9 
II 

13 
14 
16 


876 

10 8' 7 

11 10 8 

13 II 10 

14 13 II 


3.0 

.1 

.2 
•3 

•4 


0.3333 0.3322 0.331 1 0.3300 0.3289 
3226 3215 3205 3195 3185 
3125 31 15 3106 3096 3086 
3030 3021 3012 3003 2994 
2941 2933 2924 2915 2907 


0.3279 0.3268 0.3257 0.3247 0.3236 

3175 3165 3155 3145 3135 
3077 3067 3058 3049 3040 
2985 2976 2967 2959 2950 
2899 2890 2882 2874 2865 


-11 

I 
2 
3 
4 


-9 -8 -7 

1 I I 

2 2 I 

3 2 :i 

4 3 3 


3-5 
.6 

7 
.8 

•9 


0.2857 0.2849 0.2841 0.2833 0.2825 
2778 2770 2762 2755 2747 
2703 2695 2688 2681 2674 
2632 2625 2618 261 1 2604 
2564 2558 2551 2545 2538 


0.2817 0.2809 0.2801 0.2793 0.2786 
2740 2732 2725 2717 2710 
2667 2660 2653 2646 2639 
2597 2591 2584 2577 2571 
2532 2525 2519 2513 2506 


6 

7 
8 

9 
10 


5 4 4 

5 5 4 

6 6 5 
766 
8 7 6 


4.0 

.1 
.2 
•3 
•4 


0.2500 0.2494 0.2488 0.2481 0.2475 
2439 2433 2427 2421 2415 
2381 2375 2370 2364 2358 
2326 2320 2315 2309 2304 
2273 2268 2262 2257 2252 


0.2469 0.2463 0.2457 0.2451 0.2445 
2410 2404 2398 2392 2387 
2353 2347 2342 2336 2331 
2299 2294 2288 2283 2278 
2247 2242 2237 2232 2227 


-6 

I 
2 
2 


-5 -4 
I 

1 I 

2 I 
2 2 


4-S 
.6 

•7 
.8 

•9 


0.2222 0.2217 0.22120.22080.2203 
2274 2169 2165 2160 2155 
2128 2123 2119 2114 2110 
2083 2079 2075 2070 2066 
2041 2037 2033 2028 2024 


0.21980.21930.21880.21830.2179 
2151 2146 2141 2137 2132 
2105 2101 2096 2092 2088 
2062 2058 2053 2049 2045 
2020 2016 2012 2008 2004 


3 
4 
4 

5 

5 


3 2 

3 2 

4 3 

4 3 

5 4 



RECIP. 



(34) 



RECIPROCALS. 



RECIP. 



No. 


12 3 4 


56789 


interpolation! 
for thous. 1 


5.0 
.1 

.2 

•3 

•4 


0.2000 0.1996 0.1992 0.1988 0.1984 
I96I 1957 1953 1949 1946 
1923 I919 1916 I9I2 1908 
1887 1883 1880 1876 1873 
1852 1848 1845 1842 1838 


0.1980 0.1976 0.1972 0.1969 0.1965 
1942 1938 1934 1931 1927 
1905 1901 1898 1894 1890 
1869 1866 1862 1859 1855 
1835 1832 1828 1825 1821 


-A 


1 

1 
2 


-3 



I 
I 
1 


5-5 
.6 

•7 
.8 

•9 


O.1818 0.1815 O.1812 0.1808 0.1805 
1786 1783 1779 1776 1773 

1754 1751 1748 1745 1742 
1724 I72I I718 I715 I712 
1695 1692 1689 1686 1684 


0. 1 802 0. 1 799 0. 1 795 0. 1 792 0. 1 789 
1770 1767 1764 1761 1757 

1739 1736 1733 1730 1727 
1709 1706 1704 1701 1698 
1681 1678 1675 1672 1669 


2 
2 
3 
3 

4 


2 
2 
2 
2 
3 


6.0 

.1 

.2 

■3 
■4 


0.1667 0.1664 O.I66I 0.1658 0.1656 
1639 1637 1634 163I 1629 
1613 1610 1608 1605 1603 
1587 1585 1582 1580 1577 
1563 1560 1558 1555 1553 


0. 1 653 0. 1 650 0. 1 647 0. 1 645 0. 1 642 
1626 1623 1621 1618 1616 
1600 1597 1595 1592 1590 
1575 1572 1570 1567 1565 
1550 1548 1546 1543 1541 


-3 



1 
I 

I 


-2 





6-S 
.6 

■7 
.8 

•9 


0.1538 0.1536 0.1534 O.I53I 0.1529 
I515 I513 I5II 1508 1506 
1493 1490 1488 i486 1484 
1471 1468 1466 1464 1462 
1449 1447 1445 1443 1441 


0.1527 0.1524 0.1522 0.1520 0.1517 
1504 1502 1499 1497 1495 
1481 1479 1477 1475 1473 
1460 1458 1456 1453 145 I 
1439 1437 1435 1433 1431 


2 

2 
2 
2 
3 


2 
2 


7.0 

.1 

.2 

•3 
■4 


0.1429 0.1427 0.1425 0.1422 0.1420 
1408 1406 1404 1403 I4OI 
1389 1387 1385 1383 I381 
1370 1368 1366 1364 1362 
1351 1350 1348 1346 1344 


0.1418 0.1416 0.14140.1412 0.1410 
1399 1397 139s 1393 1391 
1379 1377 1376 1374 1372 
1361 1359 1357 1355 1353 
1342 1340 1339 1337 1335 


-2 





-1 







7-S 
.6 

■7 
.8 

■9 


0.1333 O.I332 0.1330 0.1328 0.1326 
1316 1314 1312 1311 1309 
1299 1297 1295 1294 1292 
1282 1280 1279 1277 1276 
1266 1264 1263 1261 1259 


0.1325 0.1323 0.1321 0.1319 0.1318 
1307 1305 1304 1302 1300 
1290 1289 1287 1285 1284 
1274 1272 1271 1269 1267 
1258 1256 1255 1253 1252 


2 
2 




8.0 

.1 

.2 

•3 

•4 


0.1250 0.1248 0.1247 0.1245 0.1244 
1235 1233 1232 1230 1229 
1220 1218 1217 1215 1214 
1205 1203 1202 1200 1199 
1190 1189 1188 1186 1185 


0.1242 0.1241 0.1239 0.1238 0.1236 
1227 1225 1224 1222 1221 
1212 1211 1209 1208 1206 
1198 1196 1195 1193 1192 
1183 1182 1181 1179 1178 


-3 











8-5 
.6 

•7 
.8 

•9 


0.11760.11750.11740.11720.1171 
1163 1161 1160 1159 1157 
H49 1148 1147 1145 1144 
1136 1135 1134 1133 1131 
1124 1122 1121 1120 1119 


0.1 170 0.1 168 0.1 167 0.1 166 0.1 164 
1156 1155 1153 1152 1151 
1143 1142 1140 1139 1138 
1130 1129 1127 1126 1125 
1117 1116 1115 1114 1112 


2 
2 




9.0 

.1 

.2 

•3 
•4 


iiii 0.11100.1109 0.H07 0.1106 
1099 1098 1096 1095 10.94 
1087 1086 1085 1083 1082 
1075 1074 1073 1072 1071 
1064 1063 1062 1060 1059 


0.1105 0.11040.11030.1101 0.1100 
1093 1092 1091 1089 1088 
ro8i 1080 1079 1078 1076 
1070 1068 1067 1066 1065 
1058 1057 1056 1055 1054 


-3 











9-S 
.6 

•7 
.8 

•9 


0.1053 0.1052 0.1050 0.1049 0.1048- 
1042 1041 1040 1038 1037 
1031 1030 1029 1028 1027 
1020 1019 1018 1017 1016 
1010 1009 1008 1007 1006 


0.1047 0.1046 0.1045 0.1044 0.1043 
1036 1035 1034 1033 1032 
1026 1025 1024 1022 1021 
1015 1014 1013 1012 1011 
1005 1004 1003 1002 1001 


2 
2 





(35) 



RECIP. 



SLIDE-WIRE RATIOS. 



S. W. RATIOS. 



cm. 


Qinm. jmm. 2^^* omm. ^mm. 


gmm. 


gmm. 


nmm. 


gmm. 


gnun. 





0.0000 0.00 10 0.0020 0.0030 0.0040 


0.0050 


0.0060 


0.0071 


0.0081 


0.0091 


I 


oioi oiii 0122 0132 0142 


0152 


0163 


0173 


0183 


0194 


2 


0204 0215 0225 0235 0246 


0256 


0267 


0278 


0288 


0299 


3 


0309 0320 0331 0341 0352 


0363 


0373 


0384 


0395 


0406 


4 


0417 0428 0438 0449 0460 


0471 


0482 


0493 


0504 


0515 


5 


0.0526 0.0537 0.0549 0.0560 0.0571 


0.0582 


0.0593 


0.0605 


0.0616 


0.0627 


6 


0638 0650 0661 0672 0684 


0695 


0707 


0718 


0730 


0741 


7 


0753 0764 0776 0788 0799 


081 1 


0823 


0834 


0846 


0858 


8 


0870 0881 0893 0905 0917 


0929 


0941 


0953 


0965 


0977 


9 


0989 looi 1013 1025 1038 


1050 


1062 


1074 


1087 


1099 


10 


o.iiii 0.1124 0.1136 0.1148 0.1161 


0.1173 


0.1186 


0.1 198 


0.1211 


0.1223 


II 


1236 1249 I26I 1274 1287 


1299 


1312 


1325 


1338 


1351 


12 


1364 1377 1390 1403 I4I6 


1429 


1442 


H55 


1468 


I48I 


13 


1494 1508, 1521 1534 1547 


1561 


1574 


1588 


1601 


I6I4 


14 


1628 1641 1655 1669 1682 


1696 


1710 


1723 


1737 


I75I 


IS 


0.1765 0.1779 0.1793 0.1806 0.1820 


0.1834 


0.1848 


0.1862 


0.1877 


0.I89I 


i6 


1905 I9I9 1933 1947 1962 


1976 


1990 


2005 


2019 


2034 


17 


2048 2063 2077 2092 2107 


2121 


2136 


2151 


2166 


2180 


i8 


2195 2210 2225 2240 2253 


2270 


2285 


2300 


2315 


2331 


19 


2346 2361 2376 2392 2407 


2422 


2438 


2453 


2469 


2484 


20 


0.2500 0.2516 0.2531 0.2547 0.2563 


0.2579 


0.2595 


0.2610 


0.2626 


0.2642 


21 


2658 2674 2690 2707 2723 


2739 


2755 


2771 


2788 


2804 


22 


2821 2837 2854 2870 2887 


2903 


2920 


2937 


2953 


2970 


23 


2987 3004 3021 3038 3055 


3072 


3089 


3106 


3123 


3I4I 


24 


3'58 3175 3193 3210 3228 


3245 


3263 


3280 


3298 


3316 


25 


0-3333 0.3351 0.3369 0.3387 0.3405 


0.3423 


0.3441 


0.3459 


0.3477 


0.3495- 


26 


3514 3532 3550 3569 3587 


3605 


3624 


3643 


3661 


3680 


27 


3699 3717 3736 3755 3774 


3793 


3812 


3831 


3850 


3870 


28 


3889 3908 3928 3947 3967 


3986 


4006 


4025 


4045 


4065 


29 


4085 4104 4124 4144 4164 


4184 


4205 


4225 


4245 


4265 


30 


0.4286 0.4306 0.4327 0.4347 0.4368 


0.4389 


0.4409 


0.4430 


0.4451 


0.4472 


3' 


4493 4SH 4535 455^ 4577 


4599 


4620 


4641 


4663 


4684 


32 


4706 4728 4749 4771 4793 


4815 


4837 


4859 


4881 


4903 


33 


4925 4948 4970 4993 5015 


5038 


5060 


5083 


5106 


5129 


34 


5152 5175 5198 5221 5244 


5267 


5291 


5314 


5337 


5361 


35 


0.5385 0.5408 0.5432 0.5456 0.5480 


0.5504 


0.5528 


0.5552 


0.5576 


0.5601 


36 


5625 5650 5674 5699 5723 


5748 


5773 


5798 


5823 


5848 


37 


5873 5898 5924 5949 5974 


6000 


6026 


6051 


6077 


6103 


38 


6129 6155 6181 6208 6234 


6260 


6287 


6313 


6340 


6367 


39 


6393 6420 6447 6475 6502 


6529 


6556 


6584 


661 1 


6639 


40 


0.6667 0-5695 0.6722 0.6750 0.6779 


0.6807 


0.6835 


0.6863 


0.6892 


0.6921 


4' 


6949 6978 7007 7036 7065 


7094 


7123 


7153 


7182 


7212 


42 


7241 7271 7301 7331 7361 


7391 


7422 


7452 


7483 


7513 


43 


7544 7575 7606 7637 7668 


7699 


7731 


7762 


7794 


7825 


44 


7857 7889 7921 7953 7986 


8018 


8051 


8083 


8116 


8149 


45 


0.8182 0.8215 0.8248 0.8282 0.8315 


0.8349 


0.8382 


0.8416 


0.8450 


0.8484 


46 


8519 8553 8587 8622 8657 


8692 


8727 


8762 


8797 


8832 


47 


8868 8904 8939 8975 901 I 


9048 


9084 


9121 


9157 


9194 


48 


9231 9268 9305 9342 9380 


9418 


9455 


9493 


9531 


9570 


49 


9608 9646 9685 9724 9763 


9802 


9841 


9881 


9920 


9960 



S. W. RATIOS. 



(36) 



SLIDE-WIRE RATIOS. 



S. W. RATIOS. 



cm. 


qUUII* 


jnun. 


2iiiin. 


3"™- 


^mm. 


gum. 


gmm. 


wmm. 


gmm. 




50 


1. 000 


1.004 


1.008 


1.012 


1.016 


1.020 


1.024 


1.028 


1-033 


1-037 


51 


1.041 


1.045 


1.049 


1-053 


1.058 


1.062 


1.066 


1.070 


1-075 


1.079 


52 


1.083 


1.088 


1.092 


1.096 


1. 101 


1.105 


1. 110 


1.114 


1.119 


1.123 


53 


I.I28 


1. 132 


I-137 


r.141 


1.146 


1.151 


1-155 


i.i6o 


1.165 


1.169 


54 


1-174 


1.179 


1-183 


1.188 


1-193 


1.198 


1-203 


1.208 


1.212 


1.217 


55 


1.222 


1.227 


1.232 


1-237 


1.242 


1-247 


1.252 


1-257 


1.262 


1.268 


56 


1'273 


1.278 


1.283 


1.288 


1.294 


1.299 


1.304 


1-309 


1-315 


1.320 


57 


1.326 


1-331 


1-336 


1-342 


1-347 


1-353 


1-358 


1.364 


1-370 


1-375 


58 


1-381 


1-387 


1.392 


1-398 


1.404 


1.410 


1-415 


1.421 


1.427 


1-433 


59 


1-439 


1-445 


1.451 


1-457 


1.463 


1.469 


1-475 


1.481 


1.488 


1.494 


60 


1.500 


1.506 


I-SI3 


1.519 


1-525 


1-532 


1-538 


1-545 


1-551 


. i;558 


6i 


1.564 


1.571 


1-577 


1.584 


1.591 


1-597 


1.604 


1.611 


1.618 


1.625 


62 


1.632 


1-639 


1.646 


1-653 


1.660 


1.667 


1-674 


1.681 


1.688 


1-695 


63 


1-703 


1.710 


1.717 


1-725 


1-732 


1.740 


1.747 


1-755 


1.762 


1.770 


64 


1.778 


1.7S6 


1-793 


1.801 


1.809 


1.817 


1.825 


1-833 


1.841 


1.849 


65 


1-857 


1.865 


1.874 


1.882 


1.890 


1.899 


1.907 


1.915 


1.924 


1-933 


66 


1. 941 


1-950 


1-959 


1.967 


1.976 


1.985 


1.994 


2.003 


2.012 


2.021 


67 


2.030 


2.040 


2.049 


2.058 


2.067 


2.077 


2.086 


2.096 


2.106 


2.115 


68 


2.125 


2-I3S 


2.145 


2-155 


2.165 


2-175 


2.185 


2.195 


2.205 


2.215 


69 


2.226 


2.236 


2.247 


2.257 


2.268 


2-279 


2.289 


2.300 


2.311 


2.322 


70 


2-333 


2.344 


2-356 


2-367 


2-378 


2.390 


2.401 


2-413 


2-425 


2.436 


71 


2.448 


2.460 


2.472 


2.484 


2.497 


2.509 


2.521 


2-534 


2.546 


2-559 


72 


2-571 


2.584 


2-597 


2.610 


2.623 


2.636 


2.650 


2.663 


2.676 


2.690 


73 


2.704 


2.717 


2-731 


2-745 


2-759 


2-774 


2.788 


2.802 


2.817 


2.831 


74 


2.846 


2.861 


2.876 


2.891 


2.906 


2.922 


2.937 


2-953 


2.968 


2.984 


75 


3.000 


3.016 


3032 


3049 


3.065 


3.082 


3.098 


3-115 


3132 


3-149 


76 


3-167 


3.184 


3.202 


3.219 


3-237 


3-255 


3-274 


3.292 


3-310 


3-329 


77 


3-348 


3-367 


3-386 


3-405 


3-425 


3-444 


3464 


3-484 


3-505 


3-525 


78 


3-545 


3-566 


3-587 


3.608 


3-630 


3-651 


3-673 


3-695 


3-717 


3-739 


79 


3.762 


3-785 


3.808 


3-831 


3-854 


3-878 


3.902 


3-926 


3-950 


3-975 


80 


4.000 


4.025 


4.051 


4.076 


4.102 


4.128 


4-155 


4.181 


4.208 


4.236 


81 


4.263 


4.291 


4-319 


4-348 


4-376 


4.405 


4-435 


4-465 


4-495 


4.525 


82 


4-556 


4-587 


4.618 


4.650 


4.682 


4-714 


4-747 


4.780 


4.814 


4.848 


83 


4.882 


4-917 


4-952 


4.988 


5.024 


5.061 


5.098 


5-135 


5-173 


5.211 


84 


5.250 


5-289 


5-329 


5-369 


5-410 


5.452 


5-494 


5-536 


5-579 


5.623 


85 


5.667 


5-7" 


5-757 


5.803 


5.849 


5-897 


5-944 


5-993 


6.042 


6.092 


86 


6.143 


6.194 


6.246 


6.299 


6-353 


6.407 


6.463 


6.519 


6-576 


6.634 


87 


6.692 


6.752 


6.813 


6.874 


6-937 


7.000 


7.065 


7-130 


7.197 


7.264 


88 


7-333 


7-403 


7-475 


7-547 


7.621 


7.696 


7.772 


7.850 


7.929 


8.009 


89 


8.091 


8.174 


8.259 


8-346 


8-434 


8.524 


8.615 


8-709 


8.804 


8.901 


90 


9.000 


9.101 


9-204 


9309 


9-417 


9-526 


9.638 


9.753 


9.870 


9.989 


91 


10. 1 1 


10.33 


10.36 


10.49 


10.63 


10-77 


10.90 


11.05 


11.20 


11-35 


92 


11.50 


11.66 


11.82 


11.99 


12.16 


12-33 


12.51 


12.70 


12.89 


13.08 


93 


13.29 


13-49 


13-71 


13-93 


14-15 


14-38 


14-63 


14.87 


15-13 


15-39 


94 


15.67 


15-95 


16.24 


16.54 


16.86 


17.18 


17-52 


17.87 


18.23 


18.61 


95 


19.00 


19.41 


19-83 


20.28 


20.74 


21.22 


21-73 


22.26 


22.81 


23-39 


96 


24.00 


24.64 


25.32 


26.03 


26.78 


27-57 


28.41 


29.30 


30.25 


31-26 


97 


32-33 


33-48 


34-71 


36-04 


37-46 


39.00 


40.67 


42.48 


44-45 


46.62 


98 


49.00 


51.6 


54-6 


57-8 


61.5 


65.7 


70.4 


75-9 


82.3 


89.9 


99 


99.0 


110. 


124. 


142. 


166. 


199- 


249. 


332- 


499- 


999. 












(37) 






s. w. 


RATIO 



NAT. SIN. 4 PL. 

NATURAL SINES AND COSINES 

TO 
FOUR PLACES. 

Note. For cosines use right-hand column of degrees and lower line of tenths. 



Deg. 


".0 ".I ".2 ".3 °.4 


°.5 °.6 °.7 °-8 ".9 




Interpola. 
for h'dth! 


0° 


0.0000 0.0017 0.0035 0.0052 0.0070 


0.0087 0.0105 0.0122 0.01400.0157 


89 


18 17 


I 


0175 0192 0209 0227 0244 


0262 0279 0297 0314 0332 


88 


2 2 


2 


0349 0366 0384 0401 0419 


0436 0454 0471 0488 0506 


87 


4 3 


3 


0523 0541 0558 0576 0593 


o6io 0628 0645 0663 0680 


86 


5 5 


4 


0698 0715 0732 0750 0767 


0785 0802 0819 0837 0854 


85 


-7 7 


S 


0.0872 0.0889 0.0906 0.0924 0.0941 


0.0958 0.0976 0.0993 0.1011*0.1028 


84 


9 9 


6 


1045 1063 1080 1097 I I 15 


1132 1149 1167 1184 1201 


83 


II 10 


7 


1219 1236 1253 1271 1288 


1305 1323 1340 1357 1374 


82 


13 12 


8 


1392 1409 1426 ii\i\/\ 1461 


1478 1495 1513 1530 1547 


81 


14 14 


9 


1564 1582 1599 1616 1633 


1650 1668 1685 1702 1719 


80° 


16 15 


10° 


0.17360.17540.17710.17880.1805 


0.1822 0.1840 0.1857 0.1874 0.1891 


79 


17 16 


II 


1908 1925 1942 1959 1977 


1994 201 I 2028 2045 2062 


78 


2 2 


12 


2079 2096 2113 2130 2147 


2164 2181 2198 2215 2232 


77 


3 3 


'3 


2250 2267 2284 2300 2317 


2334 2351 2368 2385 2402 


76 


5 5 


14 


2419 2436 2453 2470 2487 


2504 2521 2538 2554 2571 


75 


7 6 


IS 


0.2588 0.2605 0.2622 0.2639 0.2656 


0.2672 0.2689 0.2706 0.2723 0.2740 


74 


9 8 


i6 


2756 2773 2790 2807 2823 


2840 2857 2874 2890 2907 


73 


10 10 


17 


2924 2940 2957 2974 2990 


3007 3024 3040 3057 3074 


72 


12 II 


i8 


3090 3107 3123 3140 3156 


3173 3190 3206 3223 3239 


7« 


14 13 


19 


3256 3272 3289 3305 3322 


3338 3355 3371 3387 3404 


70° 


15 14 


20° 


0.3420 0.3437 0.3453 0.3469 0.3486 


0.3502 0.3518 0.3535 0.3551 0.3567 


69 


16 15 


21 


3584 3600 3616 3633 3649 


3665 3681 3697 3714 3730 


68 


2 2 


22 


3746 3762 3778 3795 381 I 


3827 3843 3859 3875 3891 


67 


3 3 


23 


3907 3923 3939 3955 397' 


3987 4003 4019 4035 4051 


66 


5 5 


24 


4067 4083 4099 4115 4131 


4147 4163 4179 4195 4210 


65 


6 6 


25 


0.4226 0.4242 0.4258 0.4274 0.4289 


0.4305 0.4321 0.4337 0.4352 0.4368 


64 


8 8 


26 


4384 4399 4415 443' 4446 


4462 4478 4493 4509 4524 


63 


10 9 


27 


4540 4555 4571 4586 4602 


4617 4633 4648 4664 4679 


62 


II II 


28 


^4695 4710 4726 4741 4756 


4772 4787 4802 4818 4833 


61 


13 12 


29 


4848 4863 4879 4894 4909 


4924 4939 4955 4970 4985 


60° 


14 14 


30° 


0.5000 0.5015 0.5030 0.5045 0.5060 


0.5075 0.5090 0.5105 0.5120 0.5135 


59 


U 13 


31 


5150 5165 5180 5195 5210 


5225 5240 5255 5270 5284 


58 


I I 


32 


5299 5314 5329 5344 5358 


5373 5388 5402 5417 5432 


57 


3 3 


33 


5446 5461 5476 5490 5505 


5519 5534 5548 5563 5577 


56 


4 4 


34 


5592 5606 5621 5635 5650 


5664 5678 5693 5707 5721 


55 


6 5 


35 


0-5736 05750 0-5764 0.5779 0-5793 


0.5807 0.5821 0.5835 0.5850 0.5864 


54 


7 7 


36 


5878 5892 5906 5920 5934 


5948 5962 5976 5990 6004 


53 


8 8 


37 


6018 6032 6046 6060 6074 


6088 6101 6115 6129 6143 


52 


10 9 


38 


6157 6170 6184 6198 6211 


6225 6239 6252 6266 6280 


51 


II 10 


39 


6293 6307 6320 6334 6347 


6361 6374 6388 6401 6414 


50° 


13 12 




i°.o ".9 °-8 °.7 °.6 


°.5 °.4 ".3 °.2 °.i 


Deg. 


Interpola. 
for h'dth! 



NAT. COS. 4 PL. 



(38) 



NATURAL SINES AND COSINES. 

4 PL. NAT. SIN. 



Oeg. 


°.o °.I °.2 ^.3 °.4 


^.5 °.6 °.7 °-8 ;.9 




Interpola. 
for h'dths 


40° 


0.6428 0.6441 0.6455 0.6468 0.6481 


0.64940.65080.6521 0.65340.6547 


49 


13 12 


41 


6561 6574 6587 6600 6613 


6626 6639 6652 6665 6678 


48 


I I 


42 


6691 6704 6717 6730 6743 


6756 6769 6782 6794 6807 


47 


3 2 


43 


6820 6833 6845 6858 6871 


6884 6896 6909 6921 6934 


46 


4 4 


44 


6947 6959 6972 6984 6997 


7009 7022 7034 7046 7059 


-45 


5 5 


45 


0.7071 0.7083 0.7096 0.7108 0.7120 


0.7133 0.7145 0.7157 0.7169 0.7181 


44 


7 6 


46 


7193 7206 7218 7230 7242 


7254 7266 7278 ,7290 7302 


43 


8 7 


47 


73*4 7325 7337 7349 736i 


7373 7385 7396 7408 7420 


42 


9 8 


48 


7431 7443 7455 7466 7478 


7490 7501 7513 7524 7536 


41 


10 10 


49 


7547 7559 757° 7S8i 7593 


7604 7615 7627 7638 7649 


40° 


12 II 


50° 


0.7660 0.7672 0.7683 0.7694 0.7705 


0.7716 0.7727 0.7738 0.7749 0.7760 


39 


11 9 


51 


7771 7782 7793 7804 7815 


7826 7837 7848 7859 7869 


38 


I I 


52 


7880 7891 7902 7912 7923 


7934 7944 7955 7965 7976 


37 


2 2 


53 


7986 7997 8007 8018 8028 


8039 8049 8059 8070 8080 


36 


3 3 


54 


.8090 8100 81H 8121 8131 


8141 8151 8161 8171 8181 


35 


4 4 


55 


0.8192 0.8202 0.821 1 0.8221 0.8231 


0.8241 0.8251 0.8261 0.8271 0.8281 


34 


6 5 


56 


8290 8300 8310 8320 8329 


8339 8348 8358 8368 8377 


33 


7 5 


57 


8387 8396 8406 84fs 8425 


8434 8443 8453 8462 8471 


32 


8 6 


58 


8480 8490 8499 8508 8517 


8526 8536 8545 8554 8563 


31 


9 7 


59 


8572 8581 8590 8599 8607 


8616 8625 8634 8643 8652 


30° 


10 8 


60° 


0.8660 0.8669 0.8678 0.8686 0.8695 


0.8704 0.8712 0.8721 0.87290.8738 


29 


8 7 


61 


8746 8755 8763 8771 8780 


8788 8796 8805 8813 8821 


28 


I I 


62 


8829 8838 8846 8854 8862 


8870 8878 8886 8894 8902 


27 


2 I 


63 


8910 8918 8926 8934 8942 


8949 8957 8965 8973 8980 


26 


2 2 


64 


8988 8996 9003 901 I 9018 


9026 9033 9041 9048 9056 


25 


3 3 


6s 


0.9063 0.9070 0.9078 0.9085 0.9092 


0.91000.91070.91140.9121 0.9128 


24 


4 4 


66 


9135 9143 915° 9157 9164 


9171 9178 9184 9191 9198 


23 


5 4 


67 


9205 9212 9219 9225 9232 


9239 9245 9252 9259 9265 


22 


6 5 


68 


9272 9278 9285 9291 9298 


9304 -9311 9317 9323 9330 


21 


6 6 


69 


9336 9342 9348 9354 9361 


9367 9373 9379 9385 939i 


20° 


7 6 


70° 


0.9397 0.9403 0.9409 0.9415 0.9421 


0.9426 0.9432 0.9438 0.9444 0.9449 


19 


6 4 


71 


9455 9461 9466 9472 9478 


9483 9489 9494 9500 9505 


18 


I 


72 


95H 9516 9521 9527 9532 


9537 9542 9548 9553 9558 


17 


I I 


73 


9563 9568 9573 9578 9583 


9588 9593 9598 9603 9608 


16 


2 I 


74 


9613 9617 9622 9627 9632 


9636 9641 9646 9650 9655 


15 


2 2 


75 


0.9659 0.9664 0.9668 0.9673 0.9677 


0.9681 0.9686 0.9690 0.9694 0.9699 


14 


3 2 


76 


9703 9707 971 1 9715 9720 


9724 9728 9732 9736 9740 


13 


4 2 


77 


9744 9748 9751 9755 9759 


9763 9767 9770 9774 9778 


12 


4 3 


78 


9781 9785 9789 9792 9796 


9799 9803 9806 9810 9813 


II 


5 3 


79 


9816 9820 9823 9826 9829 


9833 9836 9839 9842 9845 


10° 


5 4 


80° 


0.9848 0.9851 0.9854 0.9857 0.9860 


0.9863 0.9866 0.9869 0.9871 0.9874 


9 


3 2 


81 


9877 9880 9882 9885 9888 


9890 9893 9895 9898 9900 


8 





82 


9903 9905 9907 9910 9912 


9914 9917 9919 9921 9923 


7 


I 


83 


9925 9928 9930 9932 9934 


9936 9938 9940 9942 9943 


6 


I I 


84 


9945 9947 9949 995 i 9952 


9954 9956 9957 9959 9960 


5 


I I 


85 


0.9962 0.9963 0.9965 0.9966 0.9968 


0.9969 0.9971 0.9972 0.9973 0.9974 


4 


2 I 


86 


9976 9977 9978 9979 9980 


9981 9982 9983 9984 9985 


3 


2 I 


87 


9986 9987 9988 9989 9990 


9990 9991 9992 9993 9993 


2 


2 I 


88 


9994 9995 9995 999^ 999^ 


9997 9997 9997 9998 9998 


I 


2 2 


89 


9998 9999 9999 9999 9999 


1. 000 1. 000 1. 000 1. 000 1. 000 


0° 


3 2 




i°.o ".9 °.8 °.7 °-6 


°.5 °.4 -.3 °.2 °.i 


Deg. 


Interpola. 
for h'dths 



(39) 



4 PL. NAT. COS. 



NAT. TAN. 4 PL. 



NATURAL TANGENTS AND COTANGENTS 

TO 

FOUR PLACES. 

Note. For cotangents use right-hand column of degrees and lower line of tenths 



Deg. 


".0 \i ".2 °.3 °.4 


".5 °-6 °.7 °.8 ''.9 




Interpola. 
for h'dthi 


0° 


0.00000.0017 0.0035 0.0052 0.0070 


0.0087 0.0105 O.OI22 0.0140 0.0157 


89 


17 18 


I 


0175 0192 0209 0227 0244 


0262 0279 0297 0314 0332 


88 


2 2 


2 


0349 0367 0384 0402 0419 


0437 0454 0472 0489 0507 


87 


3 4 


3 


0524 0542 055^ 0577 0594 


o6i2 0629 0647 0664 0682 


86 


5 5 
7 7 


4 


0699 0717 0734 0752 0769 


0787 0805 0822 0840 0857 


85 


S 


0.0875 0.0892 0.0910 0.0928 0.0945 


0.09630.0981 0.0998 O.IOI6 0.1033 


84 


9 9 


6 


1051 1069 1086 1104 1122 


"39 1157 1175 1192 1210 


83 


10 II 


7 


1228 1246 1263 1281 1299 


1317 1334 1352 1370 1388 


82 


12 13 


8 


1405 1423 1441 1459 1477 


1495. 1512 1530 1548 1566 


81 


14 14 


9 


1584 1602 1620 1638 1655 


1673 1691 1709 1727 1745 


80° 


15 16 


10° 


0.1763 0.1781 0.1799 0.1817 0.1835 


0.18530.18710.18900.19080.1926 


79 


19 30 


II 


1944 ici2 1980 1998 2016 


2035 2053 2071 2089 2107 


78 


2 2 


12 


2126 2144 2162 2180 2199 


2217 2235 2254 2272 2290 


77 


4 4 


13 


2309 2327 2345 2364 2382 


2401 2419 2438 2456 2475 


76 


6 6 


14 


2493 2512 2530 2549 2568 


2586 2605 2623 2642 2661 


75 


8 8 


IS 


0.2679 0.2698 0.2717 0.2736 0.2754 


0.2773 0.2792 0.281 1 0.2830 0.2849 


74 


10 10 


16 


2867 2886 2905 2924 2943 


2962 2981 3000 3019 J038 


73 


II 12 


17 


3057 3076 3096 31:5 3134 


3153 3172 3191 3211 3230 


72 


13 14 


18 


3249 3269 3288 3307 3327 


3346 3365 3385 3404 3424 


71 


15 16 


19 


3443 3463 3482 3502 3522 


3541 3561 3581 3600 3620 


70° 


17 18 


20° 


0.3640 0.3659 0.3679 0.3699 0.3719 


0-3739 0.3759 0.3779 0.3799 0.3819 


69 


22 24 


21 


3839 3859 3879 3899 3919 


3939 3959 3979 4000 4020 


68 


2 2 


22 


4040 4061 4081 4101 4122 


4142 4163 4183 4204 4224 


67 


4 5 


23 


4245 4265 4286 4307 4327 


4348 4369 4390 44" 4431 


66 


7 7 


24 


4452 4473 4494 45iS 453^ 


4557 4578 4599 4621 4642 


65 


9 10 


25 


0.4663 0.4684 0.4706 0.4727 0.4748 


0.4770 0.4791 0.4813 0.4834 0.4856 


64 


11 12 


26 


4877 4899 4921 4942 4964 


4986 5008 5029 5051 5073 


63 


13 14 


27 


5095 5117 5139 5161 5184 


5206 5228 5250 5272 5295 


62 


15 17 


28 


5317 5340 5362 5384 5407 


5430 5452 5475 5498 5520 


61 


18 19 


29 


5543 5566 5589 5612 5635 


5658 5681 5704 5727 5750 


60° 


20 22 


30° 


0-5774 0-5797 0-5820 0.5844 0.5867 


0.5890 0.5914 0.5938 0.5961 0.5985 


59 


26 28 


31 


6009 6032 6056 6080 6104 


6128 6152 6176 6200 6224 


58 


3 3 


32 


6249 6273 6297 6322 6346 


6371 6395 6420 6445 6469 


57 


6 6 


33 


6494 6519 6544 6569 6594 


6619 6644 6669 6694 6720 


56 


8 8 


34 


6745 6771 6796 6822 6847 


6873 6899 6924 6950 6976 


55 


10 II 


35 


0.7002 0.7028 0.7054 0.7080 0.7107 


0.7133 0.7159 0.7186 0.7212 0.7239 


54 


13 '4 


36 


7265 7292 7319 7346 7373 


7400 7427 7454 7481 7508 


53 


16 17 


37 


7536 7563 7590 7618 7646 


7673 7701 7729 7757 7785 


52 


18 20 


38 


7813 7841 7869 7898 7926 


7954 7983 8012 8040 8069 


51 


21 22 


39 


8098 8127 8156 8185 8214 


8243 8273 8302 8332 8361 


50° 


23 25 




i°.o ".9 °.8 °7 °-6 


"■5 °.4 °.3 ".2 ".I 


Deg. 


Interpola. 
or h'dtha 



NAT. COT. 4 PL. 



(40) 



NATURAL TANGENTS AND COTANGENTS. 

4 PL. NAT. TAN. 



Deg. 


°.o ".I ".2 °.3 °.4 


°.S °.6 °.7 °-8 °.9 




Interpol a, 
forh'dths 


40° 


0.8391 0.8421 0.8451 0.8481 0.85 1 1 


0.8541 0.8571 0.8601 0.8632 0.8662 


49 


30 40 


41 


8693 8724 8754 8785 8816 


8847 8878 8910 8941 8972 


48 


3 4 


42 


9004 9036 9067 9099 9131 


9163 9195 9228 9260 9293 


47 


6 8 


43 


9325 9358 9391 9424 9457 


9490 9523 9556 9590 9623 


46 


9 12 


44 


9657 9691 9725 9759 9793 


9827 9861 9896 9930 9965 


45 


12 16 


45 


i.oooo 1.0035 1.0070 1.0105 1.0141 


1.0176 1.0212 1.0247 1.0283 1.0319 


44 ' 


15 20 


46 


0355 0392 0428 0464 0501 


0538 0575 0612 0649 0686 


43 


18 24 


47 


0724 0761 0799 0837 0875 


0913 0951 0990 1028 1067 


42 


21 28 


48 


1106 1145 1184 1224 1263 


1303 1343 1383 1423 1463 


41 


24 32 


49 


1504 1544 1585 1626 1667 


1708 1750 1792 1833 1875 


40° 


27 36 


50° 


1. 1918 1. 1960 1.2002 1.2045 i-2o88 


1.2131 1.2174 1.22l8 1.2261 1.2305 


39 


50 60 


^ 


2349 2393 2437 2482 2527 


2572 2617 2662 2708 2753 


38 


5 6 


52 


2799 2846 2892 2938 2985 


3032 3079 3127 3175 3222 


37 


10 12 


S3 


3270 3319 3367 3416 3465 


3514 3564 3613 3663 3713 


36 


15 18 


54 


3764 3814 3865 3916 3968 


4019 4071 4124 4176 4229 


35 


20 24 


55 


1.4281 1.4335 14388 1.4442 1.4496 


1.4550 1.4605 1.4659 1.4715 1.4770 


34 


25 30 


56 


4826 4882 4938 4994 5051 


5108 5166 5224 5282 5340 


33 


30 36 


57 


5399 5458 5517 5577 S637 


5697 5757 5818 5880 5941 


32 


35 42 


58 


6003 6066 6128 6191 6255 


6319 6383 6447 6512 6577 


31 


40 48 


59 


6643 6709 6775 6842 6909 


■6977 7045 7113 7182 7251 


30° 


45 54 


60° 


1.73211.73911.7461 1.7532 1.7603 


1.7675 1-7747 1.7820 1.7893 1.7966 


29 


70 80 


61 


8040 81 1 5 8190 8265 8341 


8418 8495 8572 8650 8728 


28 


7 8 


62 


8807 8887 8967 9047 9128 


9210 9292 9375 9458 9542 


27 


14 16 


63 


1.9626 1.9711 1-9797 1.9883 1.9970 


2.0057 2.0145 2.0233 2.0323 2.0413 


26 


,21 24 


64 


2.0503 2.0594 2.0686 2.0778 2.0872 


2.0965 2.10602.1155 2.1251 2.1348 


25 


28 32 


65 


2.1445 2.1543 2.1642 2.1742 2.1842 


2.1943 2.2045 2.2148 2.2251 2.2355 


24 


35 40 


66 


2460 2566 2673 2781 2889 


2998 3109 3220 3332 3445 


23 


42 48 


67 


3SS9 3673 3789 3906 4023 


4142 4262 4383 4504 4627 


22 


49 56 


68 


4751 4876 5002 5129 5257 


5386 5517 5649 5782 5916 


21 


56 64 


69. 


6051 6187 6325 6464 6605 


6746 6889 7034 7179 7326 


20° 


63 72 


70° 


2.7475 2.7625 2.7776 2.7929 2.8083 


2.8239 2.8397 2.8556 2.8716 2.8878 


19 


90 


71 


2.9042 2.9208 2.9375 2.9544 2.9714 


2.9887 3.0061 3.0237 3.0415 3.0595 


18 


9 


72 


3.0777 3.0961 3.1146 3.1334 3.1524 


3.1716 3.1910 3.2106 3.2305 3.2506 


17 


18 


73 


2709 2914 3122 3332 3544 


3759 3977 4197 4420 4646 


16 


27 


74 


4874 5105 5339 5576 5816 


6059 6305 6554 6806 7062 


15 


36 


75 


3.7321 3.7583 3.7848 3.81 18 3.8391 


3.8667 3.8947 3.9232 3.9520 3.9812 


14 


45 


76 


4.0108 4.0408 4.0713 4.1022 4.133s 


4.1653 4.1976 4.2303 4.2635 4.2972 


13 


54 


77 


4.3315 4.3662 4.4015 4.4374 4.4737 


4.5107 4.5483 4.5864 4.6252 4.6646 


12 


63 


78 


4.7046 4.7453 4.7867 4.8288 4.8716 


4.91524.9594 5.0045 5.0504 5.0970 


11 


72 


79 


5.1446 5.1929 5.2422 5.2924 5.3435 


5.3955 5-4486 5.5026 5.5578 5.6140 


10° 


81 


80° 


5.6713 5.7297 5.7894 5.8502 5.9124 


5.9758 6.0405 6.1066 6.1742 6.2432 


9 




81 


6.3138 6.3859 6.4596 6.5350 6.6122 


6.6912 6.7720 6.8548 6.9395 7-0264 ■ 


8 




82 


7.1154 7.2066 7.3002 7.3962 7.4947 


7-5958 7.6996 7.8062 7.9158 8.0285 


7 




83 


8.1443 8.2636 8.3863 8.5126 8.6427 


8.7769 8.9152 9.0579 9.2052 9.3572 


6 




84 


9.5144 9.677 9.845 10.02 10.20 


10.39 10.58 10,78 10.99 11-20 


5 




8s 


11.43 11.66 11.91 12.16 12.43 


12.71 13.00 13.30 13.62 13.95 


4 




86 


14.30 14.67 15.06 15.46 15,89 


16.35 16.83 17.34 17.89 18-46 


3 




87 


19.08 19.74 20.45 21.20 22.02 


22.90 23.86 24,90 26.03 27.27 


2 




88 


28.64 30.14 31.82 33.69 35.80 


38.19 40.92 44.07 47.74 52.08 


I 




89 


57.29 63.66 71.62 81.85 95.49 


114.6 143.2 191.0 286.5 5730 


0° 






i°.o ".9 °.8 °.7 °.6 


".5 °.4 ^.3 °.2 °.I 


Deg, 


Interpola, 
for h'dths 



(41) 



4 PL. NAT. COT. 



LOG. SIN. 4 PL. 



LOGARITHMS OF SINES AND COSINES 

TO 

FOUR PLACES. 

Note. For log. cos. use right-hand column of degrees and lower line of tenths. 



Deg. 


°.o ".I •'.7. \z °.4 


°.S °-6 °.7 °.8 °.9 




Interpola. 
for h'dths 


0° 


-00 3.24193.54293.71903.8439 


3.9408 5.0200 5.0870 5.1450 5.1961 


89 


35 25 


I 


2.2419 2.2832 2.3210 2.3558 2.3880 


2.4179 5.4459 5.4723 5.49^1 5.5206 


88 


4 3 


2 


2.5428 2.5640 2.5842 2.6035 26220 


2.6397 2.6567 2.6731 2.6889 2.7041 


87 


7 5 


3 


2.7188 2.7330 2.7468 2.7602 2.7731 


2.7857 2.7979 2.8098 2.8213 2.8326 


86 


II 8 


4 


2.8436 2.8543 2.8647 2.8749 2.8849 


2.8946 2.9042 2.9135 2.92265.9315 


85 


14 10 


S 


5.9403 5.9489 5.9573 2.9655 29736 


5.9816 2.9894 5.9970 7.0046 7.01 20 


84 


18 13 


6 


1.0192 1.0264 '•0334 1-0403 1.0472 


1 -0539 7.0605 7.0670 7.0734 7.0797 


83 


21 15 


7 


7.0859 T.0920 7.0981 7.10407.1099 


1.1157 1.12147.1271 1.13267.1381 


82 


25 18 


8 


7.1436 7.1489 7.1542 7.1594 7.1646 


1^.16971.17471.17977.18477.1895 


81 


28 20 


9 


1. 1943 1.1991 1.2038 1.2085 1^-2131 


1. 2176 1.2221 1.2266 1.2310 1.2353 


80° 


32 23 


10° 


7.2397 7.2439 7.2482 7.2524 7.2565 


7.2606 7.2647 7.2687 7-2727 7.2767 


79 


21 18 


II 


2806 2845 2883 2921 2959 


2997 3034 3070 3107 3143 


78 


2 2 


12 


3179 3214 3250 3284 3319 


3353 3387 3421 3455 3488 


77 


4 4 


'3 


3521 3554 3586 3618 3650 


3682 3713 3745 3775 3806 


76 


6 5 


"4 


3837 3867 3897 3927 3957 


3986 4015 -4044 4073 4102 


75 


8 7 


'5 


7.4130 7.4158 7.4186 7.4214 7.4242 


7.4269 7.4296 7.4323 7.4350 7.4377 


74 


II 9 


i6 


4403 4430 4456 4482 4508 


4533 4559 4584 4609 4634 


73 


13 II 


'7 


4659 4684 4709 4733 4757 


4781 4805 4829 4853 4876 


72 


15 13 


i8 


4900 4923 4946 4969 4992 


5015 5037 5060 5082 5104 


71 


17 14 


19 


5126 5148 5170 5192 5213 


5235 5256 5278 5299 5320 


70° 


19 16 


20° 


I-S34I i'.536i 7.53827.54027.5423 


7.5443 7.5463 7.5484 7.5504 7.5523 


69 


IS 13 


21 


5543 5563 5583 5602 5621 


5641 5660 5679 5698 5717 


68 


2 1 


22 


5736 5754 5773 5792 5810 


5828 5847 5865 5883 5901 


67 


3 3 


23 


5919 5937 5954 5972 599o 


6007 6024 6042 6059 6076 


66 


5 4 


24 


6093 6110 6127 6144 6161 


6177 6194 6210 6227 6243 


65 


6 5 


2S 


7.6259 7.6276 7.6292 7.6308 7.6324 


7.6340 7.6356 7.6371 7.6387 7.6403 


64 


8 7 


26 


6418 6434 6449 6465 6480 


6495 6510 6526 6541 6556 


63 


9 8 


27 


6570 6585 6600 6615 6629 


6644 6659 6673 6687 6702 


62 


II 9 


28 


6716 6730 6744 6759 6773 


6787 6801 6814 6828 6842 


61 


12 10 


29 


6856 6869 6883 6896 6910 


6923 6937 6950 6963 6977 


60° 


14 12 


30° 


7.6990 7.7003 7.7016 7.7029 7.7042 


7.7055 7.7068 7.7080 7.7093 7.7106 


59 


11 9 


31 


7118 7131 7144 7156 7168 


7181 7193 7205 7218 7230 


58 


I I 


32 


7242 7254 7266 7278 7290 


7302 7314 7326 7338 7349 


57 


2 2 


33 


7361 7373 7384 7396 7407 


7419 7430 7442 7453 7464 


56 


3 3 


34 


7476 7487 7498 7509 7520 


7S3I 7542 7553 7564 7575 


55 


4 4 


35 


7.7586 7.7597 7.7607 7.761 8 7.7629 


7.76407.76507.7661 7.7671 7.7682 


54 


6 5 


36 


7692 7703 7713 7723 7734 


7744 7754 7764 7774 7785 


S3 


7 5 


37 


7795 7805 7815 7825 7835 


7844 7854 7864 7874 7884 


52 


8 6 


38 


7893 7903 7913 7922 7932 


7941 7951 7960 7970 7979 


S> 


9 7 


39 


7989 7998 8007 8017 8026 


8035 8044 8053 8063 8072 


50° 


10 8 




i°.o °.9 °.8 °.7 °.6 


°.5 °.4 °.3 °.2 =.i 


Deg. 


Interpola. 
for h'dth! 



LOG. COS. 4 PL. 



(42) 



LOGARITHMS OF SINES AND COSINtS. 

4 PL. LOG. SIN. 



Deg. 


°.o °.i °.2 °.3 °.4 


".5 °.6 °.7 °-8 °.9 




Interpola. 
tables. 


40^ 


T.8081 T.8090T.8099T.8108T.8117 


7.8125 T.8134 7.8143 7.8152 7.8161 


49 


9 8 7 


41 


8169 8178 8187 8195 8204 


8213 8221 8230 8238 8247 


48 


1 1 I 


42 


8255 8264 8272 8280 8289 


8297 8305 8313 8322 8330 


47 


2 2 1 


43 


8338 8346 8354 8362 8370 


8378 8386 8394 8402 8410 


46 


322 


44 


8418 8426 8433 8441 8449 


8457 8464 8472 8480 8487 


45 


4 3 3 


45 


7.8495 1-8502 T.8510 1.8517 1.8525 


7.8532 7.8540 7.8547 7.8555 7.8562 


44 


5 4 4 


46 


8569 8577 8584 8591 8598 


8606 8613 8620 8627 8634 


43 


5 5 4 


47 


8641 8648 8655 8662 8669 


8676 8683 8690 8697 8704 


43 


665 


48 


871 I 8718 8724 8731 8738 


8745 8751 8758 8765 8771 


41 


766 


49 


8778 8784 8791 8797 8804 


8810 8817 8823 8830 8836 


40° 


876 


50° 


T.8843 T.8849 1-8855 'f-8862 7.8868 


7,8874 7.8880 7.8887 7.8893 7-8899 


39 


6 5 4 


SI 


8905 8911 8917 8923 8929 


8935 8941 8947 8953 8959 


38 


1 I 


52 


8965 8971 8977 8983 8989 


8995 9000 9006 9012 9018 


37 


1 I 1 


53 


9023 9029 9035 9041 9046 


9052 9057 9063 9069 9074 


36 


2 2 I 


54 


go8o 9085 9091 9096 9101 


9107 9112 9118 9123 9128 


35 


.222 


55 


7.9134 7.9139 7.9144 7.9149 7.915s 


7.91607.9165 7.9170 7.9175 7.9181 


34 


332 


56 


9186 9191 9196 9201 9206 


9211 9216 9221 9226 9231 


33 


432 


57 


9236 9241 9246 9251 925s 


9260 9265 9270 9275 9279 


32 


4 4 3 


58 


9284 9289 9294 9298 9303 


9308 9312 9317 9322 9326 


31 


5 4 3 


59 


9331 9335 9340 9344 9349 


9353 9358 9362 9367 937' 


30° 


5 5 4 


60° 


I 9375 7.9380 7.9384 7.9388 7.9393 


7-9397 7.9401 7.9406 7.9410 7.9414 


29 


4 3 2 


6i 


9418 9422 9427 9431 9435 


9439 9443 9447 945' 9455 


28 


000 


62 


9459 9463 9467 9471 9475 


9479 9483 9487 9491 9495 


27 


1 I 


63 


9499 9503 9506 9510 95 '4 


9518 9522 9525 9529 9533 


26 


1 I 1 


64 


9537 9540 9544 9548 955' 


9555 9558 9562 9566 9569 


25 


2 1 I 


65 


i'-95 73 I -95 76 1-9580 7.9583 7.9587 


7.95907.95947.9597 7.9601 7.9604 


24 


221 


66 


9607 9611 9614 9617 9621 


9624 9627 9631 9634 9637 


23 


2 2 1 


67 


9640 9643 9647 9650 9653 


9656 9659 9662 9666 9669 


22 


3 2 1 


68 


9672 9675 9678 9681 9684 


9687 9690 9693 9696 9699 


21 


322 


69 


9702 9704 9707 9710 9713 


9716 9719 9722 9724 9727 


20° 


432 


70° 


7.9730 7.9733 "'-9735 T.9738 i'.974i 


7.9743 7.9746 7.9749 7.9751 7.9754 


19 


3 2 1 


71 


9757 9759 9762 9764 97^7 


9770 9772 9775 9777 9780 


18 


000 


72 


9782 9785 9787 9789 9792 


9794 9797 9799 9801 9804 


17 


100 


73 


9806 9808 9811 9813 9815 


9817 9820 9822 9824 9826 


16 


I 1 


74 


9828 9831 9833 9835 9837 


9839 9841 9843 9845 9847 


15 


1 1 


75 


7.9849 7.9851 7.9853 7.9855 7.9857 


7.9859 7.9861 7.9863 7.9865 7.9867 


14 


2 1 1 


76 


9869 9871 9873 9875 9876 


9878 9880 9882 9884 9885 


13 


211 


77 


9887 9889 9891 9892 9894 


9896 9897 9899 9901 9902 


12 


2 1 I 


78 


9904 9906 9907 9909 9910 


9912 9913 9915 9916 9918 


11 


2 2 1 


79 


9919 9921 9922 9924 9925 


9927 9928 9929 9931 9932 


10° 


3 2 I 


80° 


7.9934 7.9935 7-9936 1-9937 1-9939 


7.9940 7.9941 7.9943 7.9944 7.9945 


9 


2 1 


81 


9946 9947 9949 9950 9951 


9952 9953 9954 9955 9956 


8 





82 


9958 9959 9960 9961 9962 


9963 9964 9965 9966 9967 


7 





83 


9968 9968 9969 9970 9971 


9972 9973 9974 9975 9975 


6 


1 


84 


9976 9977 9978 9978 9979 


9980 9981 9981 9982 9983 


5 


I 


85 


7.9983 7.9984 7.9985 7.9985 7.9986 


7.9987 7.9987 7.9988 7.9988 7.9989 


4 


I I 


86 


9989 9990 9990 9991 9991 


9992 9992 9993 9993 9994 


3 


I 1 


87 


9994 9994 9995 9995 9996 


9996 9996 9996 9997 9997 


2 


1 1 


88 ■ 


9997 9998 9998 9998 9998 


9999 9999 9999 9999 9999 


I 


2 I 


89 


9999 9999 zero zero zero 


zero zero zero zero zero 


0° 


2 I 




i°.o °.9 °.8 °.^ °.6 


°.5 °.4 ".3 "-2 ".I 


Deg. 


Interpola. 
for h'dths 



(43) 



4 PL. LOG. COS. 



LOG. TAN. 4 PL. 



LOGARITHMS OF TANGENTS AND COTANGENTS 

TO 

FOUR PLAGES. 

Note. For log. cot. use right-hand column of degrees and lower line of tenth 



Deg. 


°.o ".I °.2 °.3 °.4 


°.5 °-6 °.7 °-8 °.9 




Interpola. 
For h'dths 


0° 


-00 3.2419 3.5429 3.7190 3.8439 


3.9409 2.0200 2.0870 2.1450 2.1962 


89 


35 27 


I 


2.2419 2.2833 2.3211 2.3559 2.3881 


2.4181 2.4461 2.4725 2.49732.5208 


88 


4 3 


2 


2.5431 2.5643 2.5845 2.6038 2.6223 


2.6401 2.6571 2.67362,68942.7046 


87 


7 5 


3 


2.7194 2.7337 2.7475 2.7609 2.7739 


2.7865 2.7988 2.8107 2.8223 2.8336 


86 


II 8 


4 


,2.8446 2.8554 2.8659 2.8762 2.8862 


2.8960 2.9056 2.9150 2.9241 2.9331 


85 


14 11 


5 


2.9420 2.9506 2.9591 2.96742.9756 


2.9836 2.9915 2.9992 7.0068 7.0143 


84 


18 14 


6 


1 .02 1 6 1 .0289 1 .0360 1 .0430 1 .0499 


£.0567 1.0633 1.06997.07647.0828 


83 


21 16 


7 


T.0891 7.09547.10157.10761.1135 


1.11941.12521.13107.13677.1423 


82 


25 19 


« 8 


7.1478 7.1533 7.1587 7.1640 7.1693 


1. 1745 1.1797 1.1848 1.18987.1948 


81 


28 22 


9 


1. 1997 1.2046 1.2094 1. 2142 1. 2189 


1 .2236 1 .2282 7.2328 7.2374 7.2419 


80° 


32 24 


10° 


7.24637.25077.2551 7.25947.2637 


7.2680 7.2722 7.2764 7.2805 7.2846 


79 


25 23 


II 


2887 2927 2967 3006 3046 


3085 3123 3162 3200 3237 


78 


3 2 


12 


327s 3312 3349 3385 3422 


3458 3493 3529 3564 3599 


77 


5 5 


"3 


3634 3668 3702 3736 3770 


3804 3837 3870 3903 3935 


76 


8 7 


H 


3968 4000 4032 4064 4095 


4127 4*58 4189 4220 4250 


75 


10 9 


IS 


7.4281 7.431 1 7.4341 1.4371 7.4400 


7.4430 7.4459 7.4488 7.4517 7.4546 


74 


13 12 


i6 


4575 4603 4632 4660 4688 


47 '6 4744 477' 4799 4826 


73 


«s 14 


17 


4853 4880 4907 4934 4961 


4987 5014 5040 5066 5092 


72 


18 16 


i8 


5118 5143 5169 5195 5220 


5245 527° 5295 5320 5345 


71 


20 18 


19 


5370 5394 5419 5443 5467 


5491 5516 5539 5563 5587 


70° 


23 21 


20° 


7.5611 7.56347.56587.5681 7.5704 


^•5727 7.5750 1-5773 7.5796 7.5819 


69 


21 19 


21 


5842 5864 5887 5909 5932 


5954 5976 5998 6020 6042 


68 


2 2 


22 


6064 6086 6108 6129 6151 


6172 6194 6215 6236 6257 


67 


4 4 


23 


6279 6300 6321 6341 6362 


6383 6404 6424 6445 6465 


66 


6 6 


24 


6486 6506 6527 6547 6567 


6587 6607 6627 6647 6667 


65 


8 8 


25 


7.6687 7.6706 7.6726 7.6746 7.6765 


7.6785 7.6804 7.6824 7.6843 7.6863 


64 


II 10 


26 


6882 6901 6920 6939 6958 


6977 6996 7015 7034 7053 


63 


13 11 


27 


7072 7090 7109 7128 7146 


7165 7183 7202 7220 7238 


62 


15 13 


28 


7257 727s 7293 731 1 7330 


7348 7366 7384 7402 7420 


61 


17 15 


29 


7438 7455 7473 7491 7509 


7526 7544 7562 7579 7597 


60° 


19 17 


30° 


7.7614 7.7632 7.7649 7.7667 7.7684 


7.7701 7.7719 7.7736 7.7753 7.7771 


59 


17 15 


3> 


7788 7805 7822 7839 7856 


7873 7890 7907 7924 7941 


58 


2 2 


32 


7958 7975 7992 8008 8025 


8042 8059 8075 8092 8109 


57 


3 3 


33 


8125 8142 8158 8175 8191 


8208 8224 8241 8257 8274 


56 


5 5 


34 


8290 8306 8323 8339 8355 


8371 8388 8404 8420 8436 


55 


7 6 


35 


7.8452 7.8468 7.8484 7.8501 7.85 1 7 


7.8533 7.8549 7.8565 7.8581 7.8597 


54 


9 8 


36 


8613 8629 8644 8660 8676 


8692 8708 8724 8740 8755 


53 


10 9 


37 


8771 8787 8803 8818 8834 


8850 8865 8881 8897 8912 


52 


12 II 


38 


8928 8944 8959 8975 8990 


9006 9022 9037 9053 9068 


5' 


14 12 


39 


9084 9099 9115 9130 9146 


9161 9176 9192 9207 9223 


50° 


15 14 




i°.o ".9 °.8 °.7 ".6 


".5 °.4 ".3 ■'■2 ''.I 


Deg. 


Interpola, 
for h'dths 



LOG. COT. 4 PL. 



(44) 



LOGARITHMS OF TANGENTS AND COTANGENTS. 

4 PL. LOG. TAN. 



Deg. 


°.0 °.I °.2 ".3 °.4 


.5 .0 .7 -8 -9 




Interpola. 
for hdths 


40° 


T.9238 T.9254 T.9269 T.9284 T.9300 


T.931S 1.9330 T.9346T.9361 T.9376 


49 


15 16 


41 


1.9392 1.9407 1.9422 T.9438 7.9453 


1.9468 1.9483 1.9499 1. 95 14 1-9529 


48 


2 2 


42 


1.9544 1.9560 1.9575 I-9S90 1-9605 


1.9621 1.9636 1.9651 1.9666 1.9681 


47 


3 3 


43 


1.9697 1.9712 1.9727 1.9742 1.9757 


T.9772 T.9788 T.9803 T.9818 T.9833 


46 


5 5 


44 


1.9848 1.9864 1.9879 1.9894 1.9909 


r.9924 T.9939 T.9955 1-9970 ^-9985 


45 


6 6 


45 


0.0000 0.0015 0-0030 0.0045 0.0061 


0.00760.0091 0.01060.0121 0.0136 


44 


8 8 


46 


0152 0167 0182 0197 0212 


0228 0243 0258 0273 0288 


43 


9 10 


47 


0303 0319 0334 0349 0364 


0379 0395 0410 0425 0440 


42 


11 II 


48 


0456 0471 0486 0501 0517 


0532 0547 0562 0578 0593, 


41 


12 13 


49 ■ 


0608 0624 0639 0654 0670 


0685 0700 0716 0731 0746 


40° 


14 14 


50° 


0.0762 0.0777 O-0793 0.0808 0.0824 


0.0839 0.0854 0.0870 0.0885 0.0901 


39 


17 18 


S« 


0916 0932 0947 0963 0978 


0994 1010 1025 1041 1056 


38 


2 2 


52 


1072 1088 1 103 1 1 19 1 135 


1150 1166 1182 1197 1213 


37 


3 4 


S3 


1229 1245 1260 1276 1292 


1308 1324 1340 1356 1371 


36 


5 5 


S4 


1387 1403 1419 1435 1451 


1467 1483 1499 1516 1532 


35 


7 7 


SS 


0.1548 0.1564 0.1580 0.1596 0.1612 


0.1629 0.1645 0.1661 0.1677 0.1694 


34 


9 9 


56 


1710 1726 1743 1759 1776 


1792 1809 1825 1842 1858 


33 


lo 11 


57 


1875 1891 1908 1925 1941 


1958 1975 '992 2008 2025 


32 


12 13 


S» 


2042 2059 2076 2093 21 10 


2127 2144 2161 2178 2195 


31 


14 14 


59 


2212 2229 2247 2264 2281 


2299 2316 2333 2351 2368 


30° 


15 16 


60° 


0.2386 0.2403 0.2421 0.2438 0.2456 


0.2474 0.2491 0.2509 0.2527 0.2545 


29 


21 23 


61 


2562 2580 2598 2616 2634 


2652 2670 2689 2707 2725 


28 


2 2 


62 


2743 2762 2780 2798 2817 


2835 2854 2872 2891 2910 


27 


4 5 


63 


Z928 2947 2966 2985 3004 


3023 3042 3061 3080 3099 


26 


6 7 


64 


3118 3137 3157 3176 3196 


3215 3235 3254 3274 3294 


25 


8 9 


65 


0.3313 0.3333 0.3353 0.3373 0.3393 


0.3413 0.3433 0.3453 0.3473 0.3494 


24 


11 12 


66 


3514 3535 3555 357^ 3596 


3617 3638 3659 3679 3700 


23 


13 14 


67 


3721 3743 3764 3785 3806 


3828 3849 3871 3892 3914 


22 


15 16 


68 


3936 3958 3980 4002 4024 


4046 4068 4091 4113 4136 


21 


17 18 


69 


4158 4181 4204 4227 4250 


4273 4296 4319 4342 4366 


20° 


19 21 


70° 


0.4389 0.4413 0.4437 0.4461 0.4484 


0.4509 0.4533 0.4557 0.4581 0.4606 


19 


25 27 


71 


4630 4655 4680 4705 4730 


4755 4780 4805 4831 4857 


18 


3 3 


72 


4882 4908 4934 4960 4986 


5013 5039 5066 5093 5120 


17 


5 5 


73 


S'47 5174 5201 5229 5256 


5284 5312 5340 5368 5397 


16 


8 8 


74 


5425 5454 5483 55 "2 5541 


5570 5600 5629 5659 5689 


15 


10 11 


75 


0.5719 0.5750 0.5780 0.581 1 0.5842 


0.5873 0.5905 0.5936 0.5968 0.6000 


14 


13 14 


76 


6032 6065 6097 6130 6163 


6196 6230 6264 6298 6332 


*3 


15 i6 


77 


6366 6401 6436 6471 6507 


6542 6578 6615 6651 6688 


12 


18 19 


78 


6725 6763 6800 6838 6877 


6915 6954 6994 7033 7073 


11 


20 22 


79 


7113 7154 7195 7236 7278 


7320 7363 7406 7449 7493 


10° 


23 24 


80° 


0-7537 0.7581 0.7626 0.7672 0.7718 


0.7764 0.781 1 0.7858 0.7906 0.7954 


9 


35 45 


81 


0.8003 0.8052 0.8102 0.8152 0.8203 


0.8255 0.83070.83600.84130.8467 


8 


4 5 


82 


0.8522 0.8577 0.8633 0.8690 0.8748 


0.8806 0.8865 0.8924 0.8985 0.9046 


7 


7 9 


83 


0.91090.91720.92360.9301 0.9367 


0.9433 0.9501 0.9570 0.9640 0.971 1 


6 


11 14 


84 


0.97840.9857 0.9932 1.0008 1.0085 


1.0164 1.0244 1.0326 1.0409 1.0494 


5 


14 18 


85 


1.0580 1.6669 1-0759 1.0850 1.0944 


1.1040 1.1138 1.1238 1.1341 1.1446 


4 


18 23 


86 


1.1554 1.1664 I-I777 1-1893 I-20I2 


1.2135 1.2261 1.2391 1.2525 1.2663 


3 


21 27 


87 


1.2806 1.2954 1.3106 1.3264 1.3429 


1-3599 1-3777 1-3962 1.4155 1-4357 


2 


25 32 


88 


1.4569 1.4792 1.5027 1.5275 1.5539 


1.5819 1.6119 1.6441 1.6789 1.7167 


1 


28 36 


89 


1.7581 1.8038 1.8550 1.9130 1.9800 


2.0591 2.1561 2.28102.4571 2.7581 


0° 


32 41 




i°.o °.9 °.8 °.7 °.6 


-.5 °.4 "-3 '-2 ".1 


Deg. 


Interpola. 
for h'dths 



(45) 



4 PL. LOG. COT. 



LOG SIN, etc. 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 

TO 

FIVE PLACES. 

Note. The table gives the log of the natural value of the function, and hence the char- 
acteristic is negative when that value is fractional. The common practice of adding 10. to 
avoid the negative characteristic is not recommended. 



0° 


log cos 




0° 


log COS 




o'-i6' 

.i7'-28' 

29'-36' 

37'-43' 


0.00 000 

T.99 999 
1.99998 
1.99997 


44'-6o' 

32'-43' 
24'-3i' 
i7'-23' 


44'-49' 
5o'-54' 

S5'-59' 
60' 


1.99 996 

1-99 995 
1.99994 

1-99 993 


Ii'-i6' 
6'-io' 

I'-S' 
0' 




log sin 


89° 




log sin * 


89° 







0° 










0° 






r 


log sin 


log tan 


log cot 




/ 


log sin 


log tan 


log oot 




0' 


00" 


00 


00 


60' 


30' 


3-94 084 


3.94086 


2.05 914 


30' 


I 


4-46 373 


4-46 373 


3-53 627 


59 


31 


95508 


95510 


04490 


29 


2 


4.76476 


4.76476 


3-23 524 


58 


32 


96887 


96889 


03 III 


28 


3 


494085 


4-94 085 


3-05 915 


57 


33 


98223 


- 98 225 


01775 


27 


4 


3.06 579 


3.06 579 


2.93 421 


56 


34 


3.99 520 


3-99 522 


2.00478 


26 


s 


3.16 270 


3.16270 


2.83 730 


55 


35 


2.00 779 


2.00 781 


1.99 219 


25 


6 


24188 


24188 


75812 


54 


36 


02002 


02004 


97996 


24 


7 


30882 


30822 


69 118 


53 


37 


03192 


03194 


96806 


23 


8 


36682 


36682 


63318 


52 


38 


04350 


04353 


95647 


22 


9 


41797 


41797 


58203 


51 


39 


05478 


05481 


94519 


21 


10 


3-46 373 


346 373 


2.53 627 


50 


40 


2.06 578 


2.06 581 


1-93419 


20 


II 


50512 


50512 


49488 


49 


41 


07 650 


07653 


92347 


19 


12 


54291 


54291 


45709 


48 


42 


08696 


08 700 


91300 


18 


13 


57767 


57767 


42233 


47 


43 


09718 


09722 


90278 


17 


14 


60985 


60986 


39014 


46 


44 


10 717 


10720 


89280 


16 


15 


3.63 982 


3-63982 


2.36018 


45 


45 


2.11693 


2. 1 1 696 


1.88304 


15 


16 


66784 


66785 


33215 


44 


46 


12647 


12 651 


87349 


14 


17 


69417 


69418 


30582 


43 


47 


13581 


13585 


86415 


13 


18 


71 900 


71900 


28 100 


42 


48 


14495 


14500 


85 500 


12 


19 


74248 


74248 


25 752 


41 


49 


15 391 


15395 


84605 


II 


20 


3-76475 


3.76476 


2.23 524 


40 


50 


i.i6 268 


2.16 273 


1.83 727 


10 


21 


78594 


78595 


21405 


39 


51 


17 128 


17 133 


82867 


9 


22 


80615 


80615 


19385 


38 


52 


17971 


17976 


82024 


8 


23 


82545 


82546 


17454 


37 


53 


18798 


18804 


81 196 


7 


24 


84393 


84394 


15606 


36 


54 


19 610 


19 616 


80384 


6 


25 


3.86 166 


3.86 167 


2.13833 


35 


55 


2.20 407 


2.20413 


1-79587 


5 


26 


87870 


87871 


12 129 


34 


56 


21 189 


21 195 


78805 


4 


'I 


89509 


89510 


10490 


33 


57 


21958 


21 964 


78036 


3 


28 


91088 


91089 


08 91 1 


32 


58 


22 713 


22 720 


77280 


2 


29 


92 612 


92613 


07 387 


31 


59 


23456 


23462 


76538 


I 


30' 


3.94084 


3.94 086 


2.05 914 


30' 


60' 


2.24 186 


2.24 192 


1.75 808 


0' 




log cos 


log cot 


log tan 


r 




log oos 


log oot 


log tan 


r 



89^ 



(47) 



ftQo LOG SIN, etc. 
w 89° 



LOG SIN, etc. 1° 




2° 






/ 


log sin log tan log cot 


log cos 


log sin log tan log cot log cos 






0' 


2.24186 2.24192 1.75808 


1-99 993 


2.54282 2.54308 7.45692 7.99974 


60' 




I 


24903 24910 75090 


99 993 


54642 54669 45331 99973 


59 




2 


25 609 25 616 74 384 


99 993 


54 999 55027 44 973 99 973 


58 




3 


26304 26312 73688 


99 993 


55 354 55382 44618 99972 


57 




4 


26 988 26 996 73 004 


99992 


55705 55 734 44266 99972 


56 




5 


2.27661 2.27669 7.72331 


1.99992 


2.56054 2.56083 1.43 917 1.99 971 


55 




6 


28324 28332 71668 


99992 


56400 56429 43571 99971 


54 




7 


28977 28986 71 014 


99992 


56743 56773 43227 99970 


53 




8 


29621 29629 70371 


99992 


57084 57 114 42886 99970 


52 




9 


30 255 30 263 69 737 


99991 


57421 57452 42548 99969 


5' 




10 


230879 2.30888 1.69112 


7.99 991 


2-57 757 2.57788 7.42212 7.99969 


50 




II 


31495 31505 68495 


99991 


58089 58 121 41879 99968 


49 




12 


32103 32 112 67888 


99990 


58419 58451 41549 99968 


48 




>3 


32702 32 711 67289 


99990 


58 747 58 779 41 221 99 967 


47 




14 


33 292 33 302 66 698 


99990 


_ 59 072 59105 40895 99967 


46 




'S 


2-33875 2.33886 1.66 114 


1.99990 


2-59 395 2.59428 1.40572 1.99967 


45 




i6 


34450 34461 65539 


99989 


59715 59 749 40251 99966 


44 




'7 


35 018 35 029 64 971 


99989 


60033 60068 39932 99966 


43 




i8 


35578 35590 64410 


99989 


60349 60384 39616 99965 


42 




19 


36 131 36143 63857 


99989 


60 662. 60 698 39 302 99 964 


41 




20 


2.36678 2.36689 T.63311 


7.99 988 


2.60973 2.61009 7.38991 7.99964 


40 




21 


37217 37229 62771 


99988 


61282 61319 38681 99963 


39 




22 


37 750 37 762 62 238 


99988 


61 589 61 626 38 374 99 963 


38 




23 


38276 38289 61 711 


99987 


61894 61931 38069 99962 


37 




24 


38 796 38 809 61 191 


99987 


62 196 62 234 37 766 99 962 


36 




25 


2.39310 2.39323 T.60677 


1.99987 


2.62497 2.62535 1-37465 T.99961 


35 




26 


39818 39832 60168 


99986 


62795 62834 37166 99961 


34 




27 


40 320 40 334 59 666 


99986 


63091 63131 36869 99960 


33 




28 


40816 40830 59170 


99 986 


63385 63426 36574 99960 


32 




29 


41 307 41 321 58 679 


99985 


63678 63718 36282 2P959 


31 




30 


2.41 792 5.41 807 7.58 193 


7.99 985 


2.63968 2.64009 7.35991 7.99959 


30 




3' 


42272 42287 57713 


99985 


64256 64298 35702 99958 


. 29 




32 


42 746 42 762 57 238 


99984 


64543 64585 35415 99958 


28 




3i 


43216 43232 56768 


99984 


64827 64870 35130 99957 


27 




34 


43 680 43 696 56 304 


99984 


65110 65154 34846 99 956 


26 




35 


2.44139 2.44156 1.55844 


1.99983 


2.65391 2.65435 1.34565 1-99956 


25 




36 


44594 44 61 1 55389 


99983 


65670 65715 34285 99955 


24 




37 


45044 45061 54 939 


99983 


65947 65993 34007 99955 


23 




38 


45489 45507 54 493 


99982 


66223 66269 33731 99 954 


22 




39 


45930 45948 54052 


99982 


66497 66543 33457 99954 


21 




40 


2.46 366 2.46 385 T.53 615 


7.99 982 


2.66769 2.66816 7.33184 7.99953 


20 




4' 


46799 46817 53183 


99981 


67039 67087 32913 99952 


19 




42 


47 226 47 245 52 755 


99981 


67308 67356 32644 99952 


18 




43 


47 650 47 669 52 331 


99981 


67575 67624 32376 99951 


"7 




44 


48069 48089 51 911 


99980 


67841 67890 32110 99951 


16 




45 


5.48485 2.48505 7.51495 


1.99980 


2.68 104 2.68 154 7.31 846 7.99 950 


•15 




46 


48896 48917 51083 


99 979 


68367 68417 31583 99949 


'4. 




47 


49 304 49 325 50 675 


99 979 


68627 68678 31322 99949 


13 




48 


49 708 49 729 50 271 


99 979 


68 886 68938 31062 99948 


12 




49 


50 108 50 130 49 870 


99978 


69 144 69 196 30 804 99 948 


11 




50 


2.50504 2.50527 1.49473 


1.99978 


2.69400 2.69453 7.30547 7.99947 


10 




51 


50 897 50 920 49 080 


99 977 


69654 69708 30292 99946 


9 




52 


51 287 51 310 48690 


99 977 


69907 69962 30038 99946 


8 




53 


51673 51696 48304 


99 977 


70159 70214 29786 99945 


7 




54 


52055 52079 47921 


99976 


70409 70465 29535 99944 


6 




55 


2.52434 2.52459 1.47541 


1.99976 


2.70658 2.70714 1.29286 1.99944 


5 




56 


52810 52835 47165 


99 975 


70905 70962 29038 99943 


4 




57 


53 183 53 208 46 792 


99 975 


71 151 71208 28792 99942" 


3 




58 


53552 53578 46422 


99 974 


71395 71453 28547 99942 


2 




59 


53919 53 945 46055 


99 974 


71638 71697 28303 99941 


I 




60' 


2.54282 2.54308 1.45692 


1.99974 


2.71880 2.71940 1.28060 1.99940 


0' 






log 00s log cot log tan 


log sin 


log cos log cot log tan log sin 


> 


LO 
8 


G SI 
6°-8 


N, etc. 88° 

go '-"-' 


(4 


8) Q>;fo 









3° 










40 


10.40 

LOG SIN, etc 


f 


log sia 


log tan 


log cot 


log COS 


log sin 


log tan 


log oof 


log COS 









2.71 880 


2.71940 


1.28060 


1,99940 


2.84 358 


2.84 464 


1-15536 


T.99 894 


60 




I 


72 120 


72181 


27819 


99940 


84539 


84646 


15354 


99893 


5g 




2 


72359 


72420 


27580 


99 939 


84718 


84826 


15174 


99892 


58 




.3 


72597 


72659 


27341 


99938 


84897 


85006 


14994 


99891 


57 




4 


72834 


72896 


27104 


99938 


85075 


85185 


14815 


99891 


56 




5 


2.73069 


2-73 132 


1.26 868 


1-99 937 


2.85 252 


2.85 363 


1-14637 


1 .99 890 


55 




6 


73303 


73366 


26634 


99936 


85429 


85540 


14460 


99889 


54 




7 


73 535 


73600 


26400 


99936 


85605 


85717 


14283 


99888 


53 




8 


73767 


73832 


26168 


99 935 


85780 


ll^V' 


14107 


99887 


52 




9 


73 997 


74063 


25937 


99 934 


85955 


86069 


13931 


99886 


51 




10 


2.74 226 


2.74 292 


1.25 708 


7.99 934 


2.86 128 


2.86 243 


1-13 757 


r.99 88s 


50 




II 


74 454 


74521 


25479 


99 933 


86301 


86417 


13583 


99884 


49 




12 


74680 


74748 


25252 


99932 


86474 


86591 


13409 


99883 


48 




13 


74906 


74 974 


25026 


99932 


86645 


86763 


13237 


99882 


47 




H 


75130 


75199 


24801 


99931 


86816 


8693s 


13065 


99881 


46 




15 


2-75 353 


2.75 423 


1.24577 


1.99930 


2.86 987 


2.87 106 


1. 12 894 


r.99 880 


45 




i6 


75 575 


75645 


24355 


99929 


87156 


87277 


12 723 


99879 


44 




17 


75 795 


75867 


24133 


99929 


87325 


87447 


12553 


99879 


43 




18 


76015 


76087 


23913 


99928 


.87 494 


87616 


12384 


99878 


4» 




19 


76234 


76306 


23694 


99927 


87661 


87785 


12 215 


99877 


41 




20 


2.76451 


2.76 525 


1-23 475 


1.99 926 


2.87 829 


2-87953 


1. 12 047 


T.99 876 


40 




21 


76667 


76742 


23258 


99926 


87995 


88120 


II 880 


99875 


39 




22 


76883 


76958 


23042 


99925 


88 161 


88287 


11713 


99874 


38 




23 


77097 


77173 


22 827 


99924 


88326 


88453 


"547 


99873 


37 




24 


77310 


77387 


22 613 


99923 


88490 


88618 


11382 


99872 


36 




25 


2.77 522 


2.77 600 


1.22400 


1.99923 


2.88 654 


2.88 783 


I. II 217 


1-99871 


35 




26 


77 733 


77 81 1 


22 189 


99922 


88817 


88948 


II 052 


99870 


34 




27 


77 943 


78022 


21978 


99921 


88980 


89 in 


10889 


99869 


33 




28 


78152 


78232 


21 768 


99 920 


89 142 


89274 


10 726 


99868 


32 




29 


78360 


78441 


21559 


99920 


89304 


89437 


, 10563 


99867 


31 




30 


2.78 568 


2.78 649 


1.21351 


1-99919 


2.89 464 


2.89 598 


1. 10 402 


T.99 866 


30 




31 


78774 


78855 


21 145 


99918 


89625 


89760 


10240 


99865 


29 




32 


78 979 


79061 


20939 


99917 


89784 


89920 


10080 


99864 


28 




33 


79183 


79 266 


20734 


99917 


89943 


90080 


09 920 


99863 


27 




34 


79386 


79470 


20530 


99916 


90 102 


90240 


09 760 


99862 


26 




35 


2.79 588 


2.79 673 


1.20 327 


1-99915 


2.90 260 


2.90 399 


1.09 601 


T.99 861 


25 




36 


79789 


7987s 


20 125 


99914 


90417 


90557 


09443 


99860 


24 




37 


79990 


80076 


19924 


99913 


90574 


90715 


09 285 


99859 


23 




38 


80189 


80277 


19723 


99913 


90730 


90872 


09 128 


99858 


22 




39 


80388 


80476 


19524 


99912 


90885 


91 029 


08971 


99857 


21 




40 


2.80 585 


2.80 674 


1. 19 326 


T-99911 


2.91 040 


2.91 185 


1.08815 


1.99 856 


20 




41 


80782 


80872 


19 128 


99910 


91 195 


91340 


08660 


99855 


19 




42 


80978 


81068 


18932 


99909 


91349 


9149s 


08 505 


99854 


18 




43 


81 173 


81264 


18736 


99909 


91502 


91 650 


08350 


99853 


17 




44 


81367 


81459 


18541 


99908 


9165s 


91803 


08197 


99852 


16 




45 


2.81 560 


2.81 653 


1.18347 


1-99907 


2.91 807 


2.91 957 


1.08043 


1.99 851 


IS 




46 


81752 


81 846 


18 154 


99906 


91959 


92 no 


07 890 


99850 


14 




47 


81944 


82038 


17962 


99905 


92 no 


92 262 


07738 


99848 


13 




48 


82134 


82230 


17770 


99904 


92261 


92414 


07586 


99847 


12 




49 


82324 


82420 


17580 


99904 


92 41 1 


92565 


07435 


99 846 


II 




50 


2.82513 


2.82 610 


1-17390 


1.99903 


2.92 561 


2.92 716 


1.07 284 


T.99 845 


10 




51 


82 701 


82799 


17 201 


99902 


92 710 


92866 


07134 


99844 


9 




52 


82888 


82987 


17013 


95901 


92859 


93016 


06984 


99843 


8 




53 


83075 


83175 


16825 


99900 


93007 


93165 


06835 


99842 


7 




54 


83261 


83361 


16639 


99 899 , 


93154 


93313 


06687 


99841 


6 




55 


2.83 446 


2.83 547 


1-16453 


1.99898 


2.93 301 


2.93 462 


1.06 538 


1 .99 840 


5 




56 


.83630 


83732 


16268 


99898 


93448 


93609 


06391 


99839 


4 




57 


83813 


83916 


16084 


99897 


93 594 


93756 


06244 


99838 


3 




58 


83996 


84 100 


15900 


99896 


93740 


93903 


06097 


99837 


2 




59 


84177 


84282 


15718 


99895 


93885 


94049 


05951 


99836 


I 




60 


2.84 358 


2.84464 


1-15536 


1.99894 


2.94 030 


2.94 195 


1.05 805 


1 .99 834 









log cos 


log cot 


log tan 


log sin 


log oos 


log cot 


log tan 


log sin 


f 








m 


) 


(4 


9) 




86° 


LOG S 
85 


IN, e 
°-89° 


tc 



LOG SIN, etc. 



6= 



t 


log sin 


log tan 


log cot 


log cos 


log sin . 


log tan 


log oot 


log COS 




0' 


2.94030 


2.94 195 


1.05 805 


T.99 834 


i.oi 923 


T.02 162 


1.97838 


T.99 761 


60' 


I 


94174 


94340 


05 660 


99833 


02043 


02283 


97717 


99760 


59 


2 


94317 


94485 


05515 


99832 


02 163 


02404 


97596 


99 759 


58 


3 


94461 


94630 


05370 


99831 


02283 


02525 


97475 


99 757 


57 


4 


94603 


94 773 


05 227 


99830 


02402 


02645 


97 355 


99756 


56 


5 


2.94 746 


2:94917 


1.05 083 


1.99829 


T.02 520 


T.02 766 


1.97 234- 


1-99 755 


55 


6 


94887 


95060 


04940 


99828 


02 639 


02885 


97115 


99 753 


54 


7 


95029 


95202 


04798 


99827 


02757 


03005 


96995 


99752 


53 


8 


95170 


95 344 


04656 


99825 


02874 


03124 


96876 


99751 


52 


9 


95 310 


95486 


04514 


99824 


02992 


03242 


96758 


- 99 749 


51 


10 


2.95 450 


2.95 627 


1-04373 


T.99 823 


T.03 109 


T.03 361 


1.96639 


1.99748 


50 


II 


95589 


95767 


04233 


99822 


03226 


03479 


96521 


99 747 


49 


12 


95728 


95908 


04092 


99821 


03342 


03597 


96403 


99 745 


48 


13 


95867 


96047 


03953 


99 820 


03458 


03714 


96286 


99 744 


47 


14 


96005 


96187 


03813 


99819 


03574 


03832 


96168 


99742 


46 


'5 


2.96 143 


2.96 325 


1.03 675 


199817 


1.03690 


1.03948 


1.96052 


1 .99 741 


45 
44 


16 


96280 


96464 


03536 


99816 


03805 


04065 


95 935 


99740 


17 


96417 


96602 


03398 


99815 


03920 


04 181 


95819 


99738 


43 


18 


96553 


96739 


03 261 


99814 


04034 


04297 


95703 


99 737 


42 


19 


. 96 689 


96877 


03123 


99813 


04149 


04413 


95587 


99736 


41 


20 


2.96 825 


2.97013 


1.02987 


T.99 812 


T.04 262 


T.04 528 


1-95 472 


i^-99 734 


40 


21 


96960 


97150 


02850 


99810 


04376 


04643 


95 357 


99 733 


39 


22 


97095 


97285 


02715 


99809 


04490 


04758 


95242 


99731 


38 


23 


97229 


97421 


02579 


99808 


04603 


04873 


95127 


99730 


37 


24 


97363 


97556 


02444 


99807 


04715 


04987 


95013 


99728 


36 


25 


2.97 496 


2.97 691 


1.02 309 


1.99806 


1.04828 


1.05 lOI 


1.94899 


1.99 727 


35 


26 


97629 


97825 


02175 


99804 


04940 


05214 


94786 


99726 


34 


^7 


97762 


97 959 


02041 


99803 


05052 


05328 


94672 


99724 


33 


28 


97894 


98092 


01 908 


99802 


05 164 


05441 


94 559 


99723' 


32 


29 


98026 


98 225 


01775 


99801 


05275 


05553 


94 447 


99721 


31 


30 


2.98 157 


2.98 358 


1. 01 642 


1.99800 


1.05 386 


1 .05 666 


1-94 334 


1 .99 720 


30 


31 


98288 


98490 


01 510 


99798 


05497 


05778 


94222 


99718 


29 


32 


98419 


98622 


01378 


99797 


05 607 


05 890 


94 no 


99717 


28 


33 


98549 


9f753 


01247 


99796 


05717 


06002 


93998 


99716 


27 


34 


98679 


98884 


01 116 


99 795 


05827 


06 113 


93887 


99714 


26 


35 


2.98 808 


2.99015 


1.00985 


1-99 793 


1-05937 


1.06224 


1-93 776 


1-99713 


25 


36 


98937 


99145 


00855 


99792 


06046 


06335 


93665 


99711 


24 


37 


99066 


99275 


00725 


99791 


06155 


06445 


93 555 


99710 


23 


38 


99194 


99405 


00595 


99790 


06 264 


06556 


93 444 


99708 


22 


39 


99322 


99 534 


00466 


99788 


06372 


06666 


93 334 


99707 


21 


40 


2.99 450 


2.99 662 


1.00338 


1.99787 


T.06 481 


T.06 775 


1.93 225 


T.99 705 


20 


41 


2.99 577 


2.99 791 


1. 00 209 


99786 


06589 


06885 


93115 


99704 


19 


42 


2.99 704 


2.99919 


1. 00 081 


99785 


06696 


06994 


93006 


99702 


18 


43 


2.99 830 


1 .00 046 


0.99954 


99783 


06804 


07103 


92897 


99701 


17 


44 


2.99 956 


T.oo 174 


0.99 826 


99782 


06911 


07 211 


92789 


99699 


16 


45 


1.00082 


T.oo 301 


0.99 699 


1-99 781 


T.07 018 


T.07 320 


1.92680 


1.99698 


IS 


46 


00207 


00427 


99 573 


99780 


07124 


07428 


92572 


99696 


14 


47 


00332 


00553 


99 447 


99778 


07231 


07536 


92464 


99695 


13 


48 


00456 


00679 


99321 


99777 


07337 


07643 


92357 


99693 


12 


49 


00581 


00805 


99195 


99776 


07442 


07751 


92249 


99692 


II 


50 


T.oo 704 


T.oo 930 


0.99 070 


1.99775 


1.07548 


1.07858 


1.92 142 


1.99690 


10 


51 


00828 


01055 


98945 


99 773 


07653 


07964 


92036 


99689 


9 


52 


00951 


01 179 


98821 


99772 


07758 


08071 


91929 


99687 


8 


53 


01 074 


01303 


98697 


99771 


07863 


08177 


91823 


99686 


7 


54 


01 196 


01427 


98573 


99769 


07968 


08283 


91 717 


99684 


6 


55 


T.oi 318 


1. 01 550 


0.98 450 


1.99 768 


T.08 072 


T.08 389 


1.91 6n 


1.99683 


5 


56 


01 440 


01673 


98327 


99767 


08176 


08495 


91505 


99681 


4 


57 


01 561 


01 796 


98204 


99765 


08280 


08600 


91 400 


99680 


3 


58 


01682 


01 918 


98082 


99764 


08383 


08705 


91295 


99678 


2 


59 


01 803 


02040 


97960 


99763 


08486 


08810 


91 190 


99677 


I 


60' 


1. 01 923 


T.02 162 


0.97 838 


1.99761 


1.08589 


1 .08 9 14 


1. 9 1 086 


1.99675 


0' 




log COB 


log oot 


log tan 


log sin 


log cos 


log cot 


log tan 


log sin 


r 



LOG SIN, etc. 84° 
81°-84° 



(SO) 



83= 



8^ 



5°-8° 
LOG SIN, etc. 



1 


log sin 


ogtan 


log cot 


log cos 


log sin 


log tan 


log cot 


log cos 


■-^^^^^ 


0' 


1.08589 I. 


08914 


0.91 086 


r.99 675 


T.14356 T 


14780 


0.85 220 


1-99 575 


60' 


I 


08692 


09019 


90981 


99674 


14 445 


14872 


85 128 


99 574 


59 


2 


08 795 


09123 


90877 


99672 


14 535 


14963 


85037 


99572 


58 


3 


08 897 


09227 


90773 


99670 


14 624 


15054 


84946 


99570 


57 


4 


08999 


39330 


90670 


99669 


14714 


15 145 


84855 


99568 


56 


5 


T.09 lOI I.( 


39434 


0.90 566 


1.99667 


1.14803 1 


15236 


0.84 764 


1.99 566 


55 


6 


09 202 


39 537 


90463 


99 666 


14891 


15327 


84673 


99565 


54 


7 


09 304 


09 640 


90360 


99 664 


14 980 


15417 


84583 


99563 


53 


8 


09 405 ( 


09742 


90258 


99663 


15069 


15508 


84492 


99561 


52 


9 


09 506 ( 


39845 


90155 


99 661 


15 157 


15598 


84402 


99 559 


51 


10 


r.09 606 T.( 


39947 


0.90053 


T.99 659 


1.15245 I 


15688 


0.84312 


1-99 557 


50 


II 


09707 


10049 


?9 95i 


99658 


IS 333 


15777 


84223 


99556 


49 


12 


09807 


10 150 


89850 


99656 


15421 


15867 


84133 


99 554 


48 


13 


09907 


10252 


89748 


9965s 


15508 


15956 


84044 


99552 


47 


14 


10006 


10353 


89647 


99653 


15596 


16046 


83954 


99550 


46 


IS 


T.io 106 T. 


10454 


0.89 546 


1-99651 


1.15683 1 


16 135 


0.83 865 


1-99548 


45 


i6 


10205 


10 555 


89445 


99650 


15770 


16/224 


83776 


99546 


44 


17 


10304 


10656 


89344 


99648 


15857 


16 312 


83688 


99 545 


43 


18 


10402 


10 756 


89244 


99647 


15944 


16401 


83599 


99 543 


-42 


19 


10 501 


10856 


89144 


99645 


16030 


16489 


835" 


99 541 


41 


20 


T.IO 599 I. 


10956 


0.89 044 


T.99 643 


T.16116 T 


16577 


0.83423 


1-99 539 


40 


21 


10697 


II 056 


88944 


99642 


16203 


16665 


83335 


99 537 


39 


22 


10795 


"155 


88845 


99640 


16289 


16753 


83247 


99 535 


38 


23 


10893 


II 254 


88746 


99638 


16374 


16841 


83159 


99 533 


37 


24 


10990 


"353 


88647 


99637 


16460 


16928 


83072 


99 532 


36 


25 


T.I I 087 T. 


11452 


0.88 548 


1.99635 


T.16545 T 


17016 


0.82 984 


1.99530 


35 


26 


II 184 


"551 


88449 


99 633 


16631 


17103 


82897 


99528 


34 


27 


II 281 


1 1 649 


88351 


99632 


16 716 


17 190 


82810 


99526 


33 


28 


"377 


"747 


88253 


99630 


16801 


17277 


82723 


99524 


32 


29 


1 1 474 


11845 


881SS 


99629 


16886 


17363 


82637 


99522 


31 


30 


T.11570 I. 


"943 


0.88057 


1.99627 


T.16970 T 


17450 


0.82 550 


T.99 520 


30 


31 


II 666 


12040 


87960 


99625 


17055 


17536 


82464 


99518 


29 


32 


II 761 


12 138 


87862 


99624 


17 139 


17 622 


82378 


99517 


28 


33 


"857 . 


12235 


87765 


99622 


17223 


17708 


82292 


99515 


27 


34 


11952 


12332 


87668 


99620 


17307 


17794 


82206 


99513 


26 


35 


1. 12 047 I. 


12428 


0.87 572 


T.99 618 


1-17391 I 


17880 


0.82 120 


1-99 5" 


25 


36 


12 142 


12525 


87475 


99617 


17474 


17965 


82035 


99509 


24 


3r 


12236 


12621 


87379 


99615 


17558 


18 051 


81949 


99507 


23 


38 


12 331 


[2 717 


^7283 


99613 


17641 


18136 


81864 


99505 


22 


39 


12425 


[2813 


87187 


99612 


17724 


18221 


81779 


99503 


21 


40 


1.12519 I. 


12909 


0.87091 


T.99 610 


1.17807 I 


18306 


0.81 694 


1-99 501 


20 


41 


12 612 


13004 


86996 


99608 


17890 


18 391 


81609 


99 499 


19 


42 


12 706 


13099 


86901 


99607 


17973 


18475 


81525 


99 497 


18 


43 


12799 


13194 


86806 


99605 


18055 


18560 


81 440 


99 495 


17 


44 


12892 


13289 


86711 


99603 


18 137 


18644 


81356 


99 494 


16 


45 


T.I 2 985 T. 


13384 


0.86 616 


1.99 601 


1. 18 220 I 


18728 


0.81 272 


1.99492 


15 


46 


13078 


13478 


86522 


99600 


18 302 


18 812 


81 188 


99490 


14 


47 


13 171 


13573 


86427 


99598 


18383 


18896 


81 104 


99488 


13 


48 


13263 


13667 


86333 


99596 


18465 


18979 


81 021 


99486 


12 


49 


13355 


13761 


86239 


99 595 


18547 


19063 


80937 


99484 


II 


50 


113447 I- 


13854 


0.86 146 


I-99 593 


1. 18 628 1 


19 146 


0.80 854 


T.99 482 


10 


51 


13 539 


13948 


86052 


99 591 


18709 


19 229 


80771. 


99480 


9 


52 


13630 


14041 


85959 


99589 


18 790 


19312 


80688 


99478 


8 


53 


13722 


14134 


85866 


99588 


18871 


19395 


80605 


99476 


7 


54 


13 813 


14227 


85773 


99586 


18952 


19478 


80522 


99 474 


6 


55 


1. 13 904 I. 


14320 


0.85 680 


1.99584 


1.19033 I 


19 561 


0.80 439 


1-99472 


5 


56 


13994 


14 412 


85588 


99582 


19 113 


19643 


80357 


99470 


4 


57 


14085 


14504 


85496 


99581 


19 193 


19725 


80275 


99468 


3 


S8 


1417s 


14597 


85403 


99 579 


19273 


19807 


80193 


99 466 


2 


59 


14 266 


14688 


85312 


99 577 


19353 


19889 


80 HI 


99464 


1 


60' 


1.14356 I. 


14 780 


0.85 220 


1-99 575 


I.I9 433 I 


19971 


0.80 029 


1.99 462 


0' 




log cos 


log cot 


log tan 


log sin 


log 00s 


log cot 


log tan 


log sin 


t 



82° 



(sO 



QI0 LOG SIN, etc. 

wj. 8r-84° 



LOG SIN, etc. 



9= 



10= 



/ 


log sin 


log tan 


log oot 


log ooa 


log sin 


log tan 


log oot 


log cos 




0' 


1-19 433 


1. 19971 


3.80029 


[.99462 


T.23 967 


1.24 632 


0.75 368 


1-99 335 


60' 


I 


19513 


20053 


79 947 


99460 


24039 


24706 


75294 


99 333 


59 


2 


19592 


20134 


79866 


99458 


24 110 


24779 


75221 


99331 


58 


3 


19672 


20 216 


79784 


99456 


24 181 


24853 


75147 


99328 


57 


4 


19751 


20297 


79703 


99 454 


24253 


24926 


75074 


99326 


56 


5 


1.19830 


1.20378 


0.79 622 


1.99452 


1.24324 


T.25000 


0.75000 


1-99324 


55 


6 


19909 


20459 


79541 


99450 


24395 


25073 


74927 


99322 


54 


7 


19988 


20 540 


79460 


99448 


24466 


25 146 


74854 


99319 


53 


8 


20067 


20 621 


79 379 


99446 


24536 


25 219 


74781 


99317 


52 


9 


20145 


20 701 


79299 


99 444 


24607 


25292 


74708 


99315 


51 


10 


T.20 223 


T.20 782 


0.79 218 


7.99442 


1.24677 


T.25 365 


0.74635 


1-99313 


50 


II 


20302 


20862 


79138 


99440 


24748 


25 437 


74563 


99310 


49 


12 


20380 


20942 


79058 


99438 


24818 


25510 


74490 


99308 


48 


13 


20458 


21 022 


78978 


99436 


24888 


25582 


74418 


99306 


47 


14 


20535 


21 102 


78898 


99 434 


24958 


25655 


74 345 


99304 


46 


15 


1.20613 


7.21 182 


0.78818 


1.99432 


1.25028 


1.25 727 


0.74 273 


1-99301 


45 


i6 


20691 


21 261 


78739 


99429 


25098 


25799 


74201 


99299 


44 


17 


20768 


21341 


78659 


99427 


25 168 


25871 


74129 


99297 


43 


i8 


20845 


21 420 


78580 


99425 


25237 


25943 


74057 


99294 


42 


19 


20922 


21499 


78501 


99423 


25307 


26015 


73985 


99292 


41 


20 


T.20 999 


7,21 578 


0.78 422 


1-99 421 


1-25376 


T.26086 


0-73914 


T99 290 


40 


21 


21 076 


21657 


78343 


99419 


25 445 


26158 


73842 


99288 


39 


22 


21153 


21736 


78264 


99417 


25514 


26229 


73771 


99285 


38 


23 


21 229 


21 814 


78186 


99415 


25583 


26301 


73699 


99283 


37 


24 


21 306 


21893 


78107 


99413 


25652 


26372 


73628 


99281 


36 


25 


1.21 382 


1.21 971 


0.78 029 


1-99411 


1.25 721 


T.26 443 


0-73 557 


1.99278 


35 


26 


21458 


22049 


77951 


99409 


25790 


26514 


73486 


99276 


34 


27 


21534 


22 127 


77873 


99407 


25858 


26585 


73415 


99274 


33 


28 


21 610 


22205 


77 795 


99404 


25927 


26655 


73 345 


99271 


32 


29 


21 685 


22283 


77717 


99402 


25995 


26726 


73274 


99269 


31 


30 


7.21 761 


7.22 361 


0.77 639 


1 .99 400 


1.26063 


T.26 797 


0.73 203 


T.99 267 


30 


31 


21836 


22438 


77562 


99398 


26 131 


26867 


73133 


99264 


29 


32 


21 912 


22516 


77484 


99396 


26199 


26937 


73063 


99262 


28 


33 


21987 


22593 


77407 


99 394 


26267 


27008 


72 992 . 


99260 


27 


34 


22062 


22670 


77330 


99392 


26335 


27078 


72922 


99257 


26 


35 


7.22 137 


7.22 747 


0.77253 


1.99390 


1.26403 


1.27 148 


0.72 852 


1-99255 


25 


36 


22 211 


22824 


77176 


99388 


26470 


27 218 


72782 


99252 


24 


37 


22286 


22901 


77099 


99385 


26538 


27288 


72712 


99250 


23 


38 


22361 


22977 


77023 


99383 


26605 


27357 


72643 


99248 


22 


39 


22435 


23054 


76946 


99381 


26672 


27427 


72573 


99245 


21 


40 


7.22 509 


1.23 130 


0.76 870 


1-99 379 


7.26 739 


1.27496 


0.72 504 


T.99 243 


20 


41 


22583 


23206 


76794 


99 377 


26806 


27 566 


72434 


99241 


'i 


42 


22657 


23283 


76717 


99 375 


26873 


27635 


72365 


99238 


18 


43 


22731 


23359 


76641 


99372 


26940 


27704 


72296 


99236 


17 


44 


22805 


23435 


76565 


99370 


27007 


27773 


72227 


99233 


16 


45 


7.22 878 


1.23 510 


0.7.6 490 


1.99368 


1.27073 


1.27842 


0.72 158 


1.99231 


IS 


46 


22952 


23586 


76414 


99366 


27140 


27 911 


72089 


99229 


14 


47 


23025 


23661 


76339 


99364 


27206 


27980 


72020 


99226 


13 


48 


23098 


23737 


76263 


99362 


27273 


28049 


71 951 


99224 


12 


49 


23171 


23812 


76188 


99 359 


27339 


28117 


71883 


99221 


II 


50 


1.23244 


7.23 887 


0.76 113 


i'-99 357 


7.27 405 


T.28 186 


0.71 814 


T.99 219 


10 


51 


23317 


. 23 962 


76038 


99 355 


27471 


28254 


71746 


99217 


9 


52 


23390 


24037 


75963 


99 353 


27537 


28323 


71677 


99214 


8 


53 


23462 


24 112 


75888 


99351 


27602 


28391 


71 609 


99212 


7 


54 


23535 


24186 


75814 


99348 


27668 


28459 


71541 


99209 


6 


55 


1.23607 


7.24 261 


0.75 739 


1.99346 


1-27734 


1.28 527 


0.71 473 


1 .99 207 


5 


56 


23679 


24335 


75665 


99 344 


27799 


28595 


71405 


99204 


4 


57 


23752 


24410 


75590 


99342 


27864 


28662 


71338 


99 202 


3 


58 


23823 


24484 


75516 


99340 


27 930 


28730 


71270 


99200 


2 


59 


23895 


24558 


75442 


99 337 


27995 


28798 


71 202 


99197 


I 


60' 


1.23967 


1.24632 


0.75 368 


1-99 335 


1.28060 


T.28 865 


0.71 13s 


199195 


0' 




log G03 


log cot 


log tan 


log sin 


log 003 


log oot 


log tan 


log sin 


r 



LOG SIN, etc. 80^ 

77°-80° 



(52) 



79= 







ir 


100 ^°-12 
12 LOG SIN, 


!° 
etc 




f 


log sin log tan log cot log 00a 


log sin log tan log cot log cos 


1 




0' 

¥ 


1.28060 1.28865 0-71135 1-99 195 
28 125 28 933 71 067 99 192 
28190 29000 71000 99190 
28254 29067 70933 99187 
28319 29134 70866 99185 


1.31788 1.32747 0.67253 T.99040 


60' 


1 




A 


31 847 32 810 67 190 99 038 


59 1 




2 

3 


31907 32872 67128 99035 
31966 32933 67067 99032 


58 
57 






4 


_ 32 025 32995 67005 99030 


56 






s 


1.28384 1.29 201 0.70799 T.99182 


1.32084 1.33057 0.66943 1.99027 


55 
54 
53 
52 






6 


28448 29268 70732 99180 


32143 33 119 66881 99024 






7 


28512 2933s 70665 99177 


32 202 33 180 66 820 99 022 






8 


28577 29402 70598 99175 


32261 33242 66758 99019 






9 


28641 29468 70532 99172 


32319 33303 66697 99016 


51 






10 


1.28705 1.29535 0.70465 T.99170 


1.32378 T.33365 0.66635 1-99013 


50 






II 


28769 29601 70399 99167 


32437 33426 66574 99 on 


49 






12 


28833 29668 70332 99165 


32495 33487 66513 99008 


48 






13 


28896 29734 70266 99162 


32553 33548 66452 99005 


47 






14 


28960 29800 70200 99160 


_ 32612 33609 66391 99002 


46 






'5 


1.29024 T.29 866 0.70134 T-99 157 


1.32670 1.33670 0.66330 1.99000 


45 






i6 


29087 29932 70068 99155 


32728 33 731 66269 98997 


44 






17 


29150 29998 70002 99152 


32786 33792 66208 98994 


43 






18 


29214 30064 69936 99150 


32844 33853 66147 98991 


42 






19 


29277 30130 69870 99147 


32902 33913 66087 98989 


41 






20 


1.29340 1.30195 0.69805 T.99145 


1.32960 T.33974 0.66026 T.98986 


40 






21 


29403 30261 69739 99142 


33018 34034 65966 98983 


39 






22 


29 466 30 326 69 674 99 140 


33075 34095 65905 98980 


38 






23 


29529 30391 69609 99137 


33133 34155 65845 98978 


37 






24 


29591 30457 69543 99135 


33190 34215 65785 98975 


36 






25 


1.29654 1.30522 0.69478 1.99 132 


1-33248 1-34,276 0.65724 1.98972 


35 






26 


29716 30587 69413 99130 


33305 34336 65664 98969 


34 






^? 


29779 '30652 69348 99127 


33362 34396 65604 98967 


33 






28 


29841 30717 69283 99124 


33420 34456 65544 98964 


32 






29 


29 903 30 782 69 218 99 122 


33 477 34516 65484 98961 


31 






30 


r.29966 r. 30 846 0.69154 Y.99119 


1-33 534 T-34576 0.65424 T.98958 


30 






31 


30028 30911 69089 99 117 


33591 34635 65365 98955 


29 






32 


30090 30975 69025 99 114 


33647 34695 65305 98953 


28 






33 


30151 31040 68960 99 112 


33704 34 755 65245 98950 


27 






34 


30 213 31 104 68 896 99 109 


33761 34814 65186 98947 


26 






35 


1.30275 1.31 168 0.68832 T.99106 


1.33818 1..34874 0.65126 7.98944 


25 






36 


30336 31233 68767 99104 


33874 34 933 65067 98941 


24 






37 


30 398 31 297 68 703 99 loi 


33931 34992 65008 98938 


23 






38 


30459 31 361 68639 99099 


33987 35051 64949 98936 


22 






39 


30521 31425 68575 99096 


34043 35 III 64889 98933 


21 






40 


1.30582 T.31489 0.68 511 1.99093 


T.34 100 1.35 170 0.64 830 1.98 930 


20 






41 


30643 31552 68448 99091 


34156 35229 64771 98927 


19 






42 


30704 31 616 68384 99088 


34212 35288 64712 98924 


18 






43 


30765 31679 68321 99086 


34268 35 347 64653 98921 


17 






44 


30826 31743 68257 99083 


34324 35405 64595 98919 


16 






'^l 


1.30887 1. 3 1 806 0.68194 1.99080 


1.34380 1.35464 0.64536 1.98916 


15 






46 


30947 31870 68130 99078 


34436 35523 64477 98913 


14 






H 


31008 31933 68067 99075 


34491 35581 64419 98910 


13 






48 


31068 31996 68004 99072 


34 547 35640 64360 98907 


12 






49 


31 129 32 059 67 941 99 070 


34602 35698 64302 98904 


II 






50 


1.3 1 189 1.32 122 0.67878 T.99067 


T.34 658 1.35757 0.64243 T.98901 


10 






SI 


31250 32185 67815 99064 


34713 35815 64185 98898 


9 






52 


31310 32248 67752 99062 


34769 35873 64127 98896 


8 






53 


31370 32 311 67689 99059 


34824 35931 64069 98893 


7 






54 


_ 31 430 32373 67627 99056 


34879 35989 64011 98890 


5 






55 


1.31490 1.32436 0.67564 1.99054 


1.34934 1-36047 0.63953 1.98887 


5 






56 


31549 32498 67502 99051 


34989 36105 63895 98884 


4 






57 


31609 32561 67439 99048 


35 044 36 163 63 837 98 881 


3 






58 


31 669 32 623 67 377 99 046 


35099 36221 63779 98878 


2 






^1, 


31728 32685 67315 99-043 


_ 35 154 _ 36 279 63721 98875 


I 




. 


60' 


1.31788 1.32747 0.67253 1.99040 


1.35209 1.36336 0.63664 1.98872 


0' 




r 1 


log 00a log cot log tan log sin 


log 003 log cot log tan log sin 


r 








78° (5: 


) 77° LOG SI 


N, e1 
-80° 


tc. 



i3^-ie? 

LOG SIP), etc. 


13° 


14° 






t 


log sin 


log tan log cot log cos 


log sin log tan log cot log oob 






0' 


1.35209 


1.36336 0.63664 1.98872 


1.38368 1.39677 0.60323 T.98 690 


60' 




I 


35263 


36 394 63 606 98 869 


38418 39731 60269 98687 


59 




2 


35318 


36452 63548 98867 


38469 39785 60215 986S4 


58 




•3 


35 373 


36509 63491 98864 


38519 39838 60162 98681 


57 




4 


35427 


36 566 63 434 9-8 861 


38570 39892 60108 98678 


56 




s 


1-35481 


1.36624 0.63376 1.98858 


1.38620 1.39945 0.60055 1-98675 


55 
54 
53 
52 
51 




6 


35536 


36681 63319 98855 


38670 39999 60001 98671 




7 


35590 


36 738 63 262 98 852 


38721 40052 59948 98668 




8 


35644 


36 795 63 205 98 849 


38771 40106 59894 98665 




9 


35698 


36 852 63 148 98 846 


38821 40159 59841 98662 




10 


1 '35 752 


T.36 909 0.63091 1.98843 


1.38871 T.40212 0.59788 T.98 659 


50 

4Q 




II 


35806 


36 966 63 034 98 840 


38921 40266 59 734 98656 




12 


35860 


37023 62977 98837 


38971 40319 59.681-— TfS^ya- 


t7 

-48 




13 


35914 


37 080 62 920 98 834 


39021 40372 59628 98649_ 


47 
46 




14 


35968 


37137 62863 98831 


39071 40425 59575 98646 




IS 


1.36022 


1-37193 0.62807 1-98828 


1.39121 1.40478 0.59522 1.98643 


45 
44 
43 
42 




16 


36075 


37 250 62 750 98 825 


39170 40531 59469 98640 




17 


36129 


37 306 62 694 98 822 


39220 40584 59416 98636 




18 


36182 


37 363 62 637 98 819 


39270 40636 59364 98633 




19 


36236 


37419 62581 98816 


39319 40689 59311 98630 


41 




20 


T.36 289 


T.37476 0.62524 T.98813 


1-39369 T.40742 0.59258 T.98 627 


40 




21 


36342 


37532 62468 98810 


39418 40795 59205 98623 


39 




22 


36395 


37588 62412 98807 


39467 40847 59153 98620 


^8 




23 


36449 


37 644 62 356 98 804 


39517 40900 59100 98617 


36 




24 


36502 


37 700 62 300 98 8oi 


39566 40952 59048 98614 




25 


1-36555 


1-37756 0.62244 1.98798 


1-39 615 I-4J005 0.58995 T.98 610 


35 
34 




26 


36608 


37 812 62 188 98 795 


39664 41057 58943 98607 




27 


36660 


37 868 62 132 98 792 


39 7«3 41109 58891 98604 


33 




28 


^Vll 


37 924 62 076 98 789 


39 762 41 161 58 839 98 601 


32 




29 


36766 


37 980 62 020 98 786 


39811 41214 58786 98597 


31 




30 


r.36 819 


T.38035 0.61965 r.98783 


T.39 860 T.41 266 0.58 734 T.98 594 


30 




31 


36871 


38 091 61 909 98 780 


39909 41 318 58682 98591 


29 




32 


36924 


38 147 61 853 98 777 


39958 41370 58630 98588 


28 




33 


36976 


38 202 61 798 98 774 


40006 41422 58578 98584 


27 




34 


37028 


38 257 61 743 98 771 


40055 41474 58526 98581 


26 




35 


T.37081 


1.38 313 0.61687 1-98768 


1.40103 1.41526 0.58474 1.98578 


25 




36 


37133 


38 368 61 632 98 765 


40152 41578 58422 98574 


24 




37 


37185 


38423 61577 98762 


40200 41629 58371 98571 


23 




38 


37237 


38479 61 521 98759 


40 249 41 681 58 319 98 568 


22 




39 


37289 


38 534 6i 466 98 756 


40 297 41 733 58 267 98 565 


21 




40 


1-37 341 


1.38589 0.61 411 T.98753 


T.40346 T.41 784 0.58216 T.98 561 


20 




41 


37 393 


38 644 61 356 98 750 


40394 41836 58164 98558 


19 




42 


37 445 


38 699 ^61 301 98 746 


40442 41887 58113 98555 


18 




43 


37 497 


38 754 61 246 98 743 


40490 41939 58061 98551 


17 




44 


37 549 


38 808 61 192 98 740 


40538 41990 58010 98548 


16 




45 


1.37600 


T.38 863 0.61 137 T.98 737 


T.40586 T.42041 0.57959 T.98 545 


15 




46 


37652 


38918 61082 98734 


40634 42093 57907 98541 


14 




47 


37703 


38972 61028 98731 


40 682 42 144 57 856 98 538 


13 




48 


37 755 


39027 60973 98728 


40730 42195 57805 98535 


12 




49 


37806 


39 082 60 918 98 725 


40778 42246 57754 98531 


11 




50 


7-37 858 


T.39 136 0.60864 1.98722 


1.40825 1.42297 0.57703 1.98528 


10 




51 


37909 


39 190 60 810 98 719 


40873 42348 57652 98525 


9 




52 


37960 


39 245 60 755 98 715 


40921 42399 57601 98521 


8 




53 


38 01 1 


39299 60701 98712 


40968 42450 57550 98518 


7 




54 


38062 


39 353 60 647 98 709 


41016 42501 57499 98515 


6 




55 


T-38113 


1.39407 0.60593 1.98706 


1.41063 1.42552 0.57448 1.98511 


5 




56 


38164 


39 461 60 539 98 703 


41111 42603 57397 98508 


4 




57 


38215 


39515 60485 98700 


41 158 42 653 57 347 98 505 


3 




58 


38266 


39569 60431 98697 


41 205 42 704 57 296 98 501 


2 




59 


38317 


39 623 60 377 98 694 


41252 42755 57245 98498 


1 




60 


1.38368 


1-39677 0.60323 1.98690 


1.41300 1.42805 0.57195 1.98494 


0' 






log cos 


log cot log tan log sin 


log cos log cot log tan log sin 


1 



LOG SIN. etc. 76= 



(54) 



76= 





15° 


^^0 13°-16° 
lb LOG SIN, etc 


t 


log Bin log tan logoot log 00a 


log sin log tan logoot logoos 






0' 

I 

2 


1.41300 1.42805 0.57195 1.98494 
41347 42856 57144 98491 
41394 42906 57094 98488 


1.44034 1.45750 0.54250 1.98284 
44 078 45 797 54 203 98 281 
44122 45845 54155 98277 


60' 

59 
58 




3 


41441 42957 57043 98484 


44 166 45 892 54 108 98 273 


% 




4 


41488 43007 56993 98481 


44210 45940 54060 98270 




5 


141 535 1-43057 0.56943 1.98477 


1-44253 1.45987 0.54013 T.98 266 


55 
54 




6 


41582 43108 56892 98474 
41628 43158 56842 98471 


44297 46035 53965 98262 




7 


44341 46082 53918 98259 


S3 
52 




8 


41675 43208 56792 98467 


44385 46130 53870 98255 




9 


41 722 43 258 56 742 98 464 


44428 46177 53823 98251 


51 




10 


1.41 768 T.43 308 0.56 692 T.98 460 


1.44472 1.46224 0.53776 T.98 248 


50 




II 


41 815 43358 56642 98457 


4451& 46271 53729 982^^ 


49 




12 


41861 43408 56592 98453 


44 559 46319 53681 98240 


48 




13 


41908 43458 56542 98450 


44602 46366 53634 98237 


47 




14 


41954 43508 56492 98447 


44646 46413 53587 98233 


46 




'S 


1.42 001 1.43558 0.56442 1.98443 


1.44689 1.46460 0.53540 1.98229 


45 




i6 


42047 43607 56393 98440 


44 733 46507 53 493 98226 


44 




17 


,42093 43657 56343 98436 


44776 46554 53446 98222 


43 




18 


42140 43707 56293 98433 


44819 46601 53399 98218 


42 




19 


42186 43756 56244 98429 


44862 46648 53352 98215 


41 




20 


1.42232 T.43 806 0.56194 T.98 426 


T.44905 T.46694 0.53306 T.98 211 


40 




21 


42278 43855 56145 98422 


44948 46741 53259 98207 


39 




22 


42324 43905 56095 98419 


44992 46788 53212 98204 


38 




23 


42370 43954 56046 98415 


45035 46835 53165 98200 


37 




24 


42416 44004 55996 98412 


_ 45 077 46881 53 119 98196 


36 




^5 


1.42 461 T.44053 0.55947 T.98 409 


1.45 120 1.46928 0.53072 1.98192 


35 




26 


42507 44102 55898 98405 


45163 46975 53025 98189 


34 




27 


42553 44 151 55849 98402 


45206 47021 52979 98185 


33 




28 


42599 44 20I 55 799 98398 


45249 47068 52932 98 181 


32 




29 


42644 44250 55750 98395 


45292 47 114 52886 98177 


31 




30 


1.42690 T.44299 0.55701 T.98 391 


1-45 334 T.47160 0.52840 T.98 174 


30 




31 


42735 44348 55652 98388 


45 377 47207 52793 98170 


29 




32 


• 42781 44 397 55603 98384 


45419 47253 52747 98166 


28 




33 


42826 44446 55554 98381 


45 462 47 299 52 701 98 162 


27 




34 


42872 44495 55505 98377 


45504 47346 52654 98159 


26 




3| 


1.42 917 1-44 544 0.55456 1.98373 


1.45547 1.47392 0.52608 1.98 155 


25 




36 


42962 44592 55408 98370 


45589 47438 52562 98151 


24 




37 


43008 44641 55359 98366 


45632 47484 52516 98147 


23 




38 


43053 44690 55310 98363 


45674 47530 52470 98144 


22 




39 


43098 44738 55262 98359 


45716 47576 52424 98140 


21 




40 


1-43143 7-44787 0.55213 T.98 356 


1-45758 T.47622 0.52378 T.98 136 


20 




41 


43188 44836 55164 98352 


45801 47668 52332 98132 


19 




42 


43233 44884 55 "6 98349 


45843 47714 52286 98129 


18 




43 


43278 44 933 55067 98345 


45885 47760 52240 98125 


17 




44 


43323 44981 55019 98342 


_ 45 927 47806 52194 98121 


16 




45 


1.43367 1.45029 0.54971 1.98338 


1.45969 1.47852 0.52148 1.98 117 


15 




46 


43 4*2 45078 54922 98334 


46 01 1 47897 52103 98 113 


14 




47 


43 457 45126 54874 98331 


46053 47943 52057 98 no 


13 




48 


43502 45174 54826 98327 


46095 47989 52 01 1 98106 


12 




49 


43546-- 45222 54778 98324 


46136 48035 51965 98102 


II 




50 


1.43591 1.45 271 0.54729 T.98 320 


T.46178 T.48080 0.51920 T.98 098 


10 




51 


43635 45319 54681 98317 


46220 48126 51874 98094 


9 




52 


43680 45367 54633 98313 


46262 48 171 51829 98090 


8 




S3 


43 724 45 415 54 585 • 98 309 


46303 48217 51783 98087 


7 




54 


43769 45463 54 537 98306 


46345 48262 51738 98083 


6 




^\ 


1-43813 1.45 5" 0.54489 1.98302 


T.46386 T.48307 0.51693 T.98 079 


5 




56 


43857 45 559 54441 98299 


46428 48353 51647 98075 


4 




5^ 


43901 45606 54 394 98295 


46469 48398 51602 98071 


3 




58 


43946 45654 54346 98291 


46511 48443 51557 98067 


2 




59 


43 990 45 702 54 298 98 288 


46552 48489 51511 98063 


I 




60' 


1.44034 1.45750 0.54250 1.98284 


1.46594 1.48534 0.51466 1.98060 


0' 






log cos log cot log tan log sin 


log COS log cot log tan log sin 


t 






74 (5- 


730 LOG^Sl 


N, e1 
'-76° 


tc. 





1 7°-20° 








LOG SIN, etc. 


17° 


18° 






r 


log sin 


log tan log cot log ooa 


log sin log tan log cot log cos 






0' 


1.46 594 


1.48534 0.51466 1.98060 


1.48998 1.5 1 178 0.48822 7.97821 


60' 




I 


46635 


48579 51 421 98056 


49037 51221 48779 97817 


59 




2 


46 676 


48624 51376 98052 


49076 51264 48736 97812 


58 




3 


46717 


48669 51 331 98048 


49 115 51306 48694 97808 


57 




4 


46758 


48 714 51 286 98044 


49153 51349 48651 97804 


56 




s 


1.46800 


7.48759 0.51 241 7.98040 


1.49 192 1.51392 0.48608 1.97800 


55 
54 
53 
52 
51 




6 


46841 


48804 51 196 98036 


49231 51435 48565 97796 




7 


46882 


48849 51151 98032 


49269 51478 48522 97792 




8 


46923 


48894 51 106 98029 


49308 51520 48480 97788 




9 


46964 


48939 51061 98025 


49 347 51563 48437 97784 




10 


7.47 005 


7.48984 0.51 016 7.98021 


1.49385 1.51606 0.48394 7.97779 


50 




II 


47045 


49029 50971 98017 


49424 51648 48352 97775 


49 




12 


47086 


49073 50927 98013 


49462 51 691 48309 97771 


48 




'3 


47127 


49 "8 50882 98009 


49500 51734 48266 97767 


47 
46 




H 


47168 


49163 50837 98005 


49 539 51776 48224 97763 




'5 


1.47209 


1.49207 0.50793 1.98001 


'•49 577 1.51819 0.48181 7.97759 


45 
44 




i6 


47249 


49 252 50 748 97 997 


49615 51 861 48139 97754 




17 


47290 


49 296 50 704 97 993 


49654 51903 48097 97750 




18 


47330 


49 341 50 659 97 989 


49692 51946 48054 97746 


42 




19 


47371 


49385 50615 97986 


49730 51988 48012 97742 


41 




20 


1.47 41 1 


7.49430 0.50570 7.97982 


T.49768 7.52031 0.47969 7.97738 


40 




21 


47452 


49 474 50 526 97 978 


49806 52073 47927 97734 


39 




22 


47492 


49519 50481 97 974 


49844 52115 47885 97729 


38 




23 


47 533 


49563 50437 97970 


49882 52157 47843 97725 


37 




24 


47 573 


49 607 50 393 97 966 


49920 52200 47800 97721 


36 




^1 


1-47613 


1.49652 0.50348 1.97962 


1.49958 1.52242 0.47758 1.97 717 


35 
34 




26 


47654 


49 696 50 304 97 958 


49996 52284 47716 97713 




27 


47694 


49 740 50 260 97 954 


50034 52326 47674 97708 


33 




28 


47 734 


49 784 50 216 97 950 


50072 52368 47632 97704 


32 




29 


47 774 


49828 50172 97946 


50110 52410 47590 97700 


31 




30 


T.47814 


7.49872 0.50128 1.97942 


7.50148 1.52452 0.47548 7.97696 


30 


' 


3> 


47854 


49916 50084 97938 


50185 52494 47506 97691 


29 




32 


47894 


49 960 50 040 97 934 


50223 52536 47464 97687 


28 • 




33 


47 934 


50004 49996 97930 


50261 52578 47422 97683 


27 




34 


47 974 


50 048 49 952 97 926 


50298 52620 47380 97679 


26 




35 


1.48 014 


7.50092 0.49908 7.97922 


1.50336 1.52 661 0.47339 1.97674 


25 




36 


48054 


50 136 49 864 97 918 


50374 52703 47297 97670 


24 




^2 


48094 


50 180 49 820 97 914 


50411 52745 4725s 97666 


23 




38 


48133 


50 223 49 777 97 910 


50449 52787 47213 97662 


22 




39 


48173 


50 267 49 733 97 906 


50486 52829 47 171 97657 


21 




40 


T.48 213 


7.50 31 1 0.49689 7.97902 


7.50523 7.52870 0.47130 7.97653 


20 




41 


48252 


50 355 49 645 97 898 


50561 52912 47088 97649 


19 




42 


48292 


50 398 49 602 97 894 


50598 52953 47047 97645 


18 




43 


48332 


50442 49558 97890 


50635 52995 47005 97640 


"7 




44 


48371 


50485 49 5 '5 97886 


50673 53037 46963 97636 


16 




45 


1.48 41 1 


1.50529 0.49471 1.97882 


1.50 7 10 1.53078 0.46922 1.97632 


IS 




46 


48450 


50572 49428 97878 


50 747 53 120 46 880 97 628 


14 




47 


48490 


50616 49384 97874 


50784 53 161 46839 97623 


13 




48 


48529 


50659 49341 97870 


50821 53202 46798 97619 


12 




49 


48568 


50 703 49 297 97 866 


50858 53244 46756 97615 


. II 




50 


7.48 607 


7.50 746" 0.49 254 7.97 861 


7.50896 1.53285 0.46715 1.97 610 


10 




5« 


48647 


50789 49 21 1 97857 


50933 53327 46673 97606 


9 




52 


48686 


50833 49167 97853 


50 970 53 368 46 632 97 602 


8 




53 


48725 


50876 49 124 97849 


51007 -53409 46591 97 597 


7 




54 


48764 


50919 49081 97845 


51043 53450 46550 97593 


6 




55 


1.48803 


7.50 962 0.49 038 7.97 841 


1. 5 1 080 1.53492 0.46508 1.97589 


5 




56 


48842 


51005 48995 97837 


5H17 53 533 46467 97584 


4 




57 


48881 


51048 48952 97833 


5« 154 53 574 46426 97580 


3 




58 


48 920 


51092 48908 97829 


51 191 53615 46385 97576 


2 


59 1 


48959 


51 135 48 865 97 825 


51227 53656 46344 97571 


I 


LO 
6 


60' 


1.48998 


[.51 178 0.48822 1.97 821 


1-51264 1.53697 0.46303 1.97567 


0' 




log cos 


log cot log tan log sin 


log cos log oot log tan log sin 


> 


G SI 
9°-7: 


M, etc. 

1° 


72° (5^ 


710 








19° 




»U LOG SIN, etc 


f 


logsm log tan log cot log 00s 


log sin 


log tan log oot log 00a 






0' 


1.51264 1.53697 0.46303 1.97567 


1-53405 


1.56 107 0.43893 r.97299 


60' 




I 


5' 301 53738 46262 97563 
51338 53 779 46221 97558. 


53440 


56 146 43 854 97 294 


59 




2 


53 475 


56185 43815 97289 


58 




3 


51374 53820 46180 97 554 
SI 411 53861 46139 97550 


53509 


56224 43776 97285 


57 




4 


53 544 


56 264 43 736 97 280 


56 




6 


1-51447 1-53 902 0.46098 1.97545 
51484 53943 46057 97541 


1-53578 


1.56303 0.43697 1.97276 


55 




S3 613 


56 342 43 658 97 271 


';4 




7 


51520 53984 46016 97536 


53647 


56381 43619 97266 


S3 




8 


51557 54025 45975 97532 


53682 


56 420 43 580 97 262 


52 




9 


5' 593 54065 45935 97528 


53716 


56459 43541 97257 


51 




10 


1.51629 T.54106 0.45894 T.97523 


1-53 751 


1.56498 0.43502 1.97252 


50 




II 


51666 54147 45853 97519 


53785 


56 537 43 463 97 248 


49 




12 


51702 54187 45813 97515 


53819 


56576 43424 97243 


48 




13 


51738 54228 45772 97510 


53854 


56 615 43 385 97 238 


47 




14 


51774 54269 45731 97506 


53888 


56 654 43 346 97 234 


46 




'1 


1.51811 1.54309 0.45691 1.97 501 


1-53922 


1.56693 0.43307 1.97229 


45 




16 


51847 54350 45650 97 497 


S3 957 


56 732 43 268 97 224 


44 




17 


51883 54390 45610 97492 


53991 


56 771 43 229 97 220 


43 




18 


51919 54431 45569 97488 


. 54025 


56810 43190 97215 


42 




19 


51955 54471 45529 97484 


54059 


56849 43 151 97210 


41 




20 


T.51991 T.54512 0.45488 T.97479 


1-54093 


T.56887 0.43113 T.97206 


40 




21 


52027 54552 45448 97475 


54127 


56 926 43 074 97 201 


39 




22 


52063 54593 45407 97470 


54161 


56965 43035 97196 


38 




23 


52099 54633 45367 97466 


54195 


57 004 42 996 97 192 


37 




24 


52135 54673 45327 97461 


54229 


_ 57 042 42958 97187 


36 




25 


1.52 171 1.54714 0.45286 1.97457 


1-54263 


1.57081 0.42919 1.97182 


35 




26 


52207 54754 45246 97453 


54297 


57120 42880 97178 


34 




27 


52242 54794 45206 97448 


54 331 


57158 42842 97173 


33 




28 


52278 54835 45165 97444 


54365 


57197 42803 97168 


32 




29 


52314 5487s 45125 97439 


54 399 


57 23s 42 765 97 163 


31 




30 


1-52350 1-54915 0.45085 T.97435 


1-54 433 


1.57274 0.42726 1.97159 


30 




31 


52385 54955 45045 97430 


54466 


57312 42688 97154 


29 




32 


52421 54 995 45005 97426 


54500 


57351 42649 97149 


28 




33 


52456 55035 44965 97421 


54 534 


57389 42611 97145 


27 




34 


52492 55075 44925 97417 


54567 


57428 42572 97140 


26 




^l 


1.52527 1.55 115 0.44885 1.97412 


1.54601 


1.57466 0.42534 1.97 135 


25 




36 


52563 55155 44845 97408 


54635 


57504 42496 97130 


24 




37 


52598 55195 44805 97403 


54668 


57 543 42457 97126 


23 




38 


52634 55235 44765 97 399 


54702 


57581 42419 97121 


22 




39 


52669 55275 44725 97394 


54 735 


57619 42381 97116 


21 




40 


1.52705 1.55315 0.44685 T.97390 


1.54 769 


T.57658 0.42342 T.97111 


20 




41 


52740 55 355 44645 97385 


54802 


57 696 42 304 97 107 


19 




42 


52775 55 395 44605 973S1 


54836 


57 734 42 266 97 102 


18 




43 


52811 55434 44566 97376 


54869 


57772- 42228 97097 


17 




44 


52846 55474 44526 97372 


54903 


57810 42190 97092 


16 




45 


1.52 881 1.55514 0.44486 1.97367 


1-54936 


r.57849 0.42151 T.97087 


15 




46 


52916 55554 44446 97363 


54969 


57887 42113 97083 


14 




47 


52951 55 593 44407 97358 


55003 


57925 42075 97078 


13 




48 


52986 55633 44367 97 353 


55036 


57963 42037 97073 


12 




49 


53021 55673 44327 97349 


55069 


58 001 41 999 97 068 


II 




50 


1.53056 7.55712 0.44288 T.97344 


1.55 102 


T.58039 0.41961 1.97063 


10 




SI 


53092 55752 44248 97340 


55136 


58077 41923 97059 


9 




52 


53126 55 791 44209 97335 


55169 


58115 41885 97054 


8 




53 


53161 55831 44169 97331 


55202 


58 153 41 847 97 049 


7 




54 


53196 55870 44130 97326 


_ 55 23s 


58 191 41 809 97044 


6 




55 


1-53 231 1.55910 0.44090 1.97322 


1.55 268 


1.58229 0.41 771 1.97039 


5 




56 


53266 55949 44051 97317 


55 301 


58 267 41 733 97 035 


4 




H 


53301 55989 44 01 1 97312 


55 334 


58 304 41 696 97030 


3 




58 


53336 56028 43972 97308 


55367 


58 342 41 658 97 025 


2 




59 


_ 53 370 56067 43933 97303 


55400 


58 380 41 620 97 020 


1 




60 


1-53405 1-56107 0.43893 1.97299 


I -55 433 


1.58418 0.41582 1.97015 


0' 






log 00s log oot log tan log sin 


log 00s 


log oot log tan log sin 


1 






70° (s 


?) 


69° LOG s 
" 69 


N, el 
°-72° 


tc. 



2 1 °-24° 
LOG SIN, etc. 2r 



22° 



t 


log aiu 


log tan 


log oot 


log COS 


log sin 


log tan 


log oot 


log cos 




0' 


> -55 433 


f. 58 418 


0.41 582 


1-97015 


1-57 358 


T.60 641 


0.39 359 


T.96 717 


60' 


I 


55466 


58 455 


41545 


97010 


57389 


60 677 


39323 


96 711 


59 


2 


55 499 


58493 


41507 


97005 


57420 


60714 


39286 


96 706 


58 


3 


55532 


58531 


41469 


97001 


57451 


60 750 


39250 


96 701 


57 


4 


55564 


58569 


41431 


96996 


57482 


60 786 


39214 


96696 


56 


s 


1-55 597 


1.58606 


0.41 394 


T.96 991 


1-57 514 


T.60 823 


0.39 177 


T.96 691 


55- 


6 


55630 


58644 


41356 


96986 


57 545 


60859 


39 141 


96686 


54 


7 


55663 


58681 


41 319 


96981 


57576 


60895 


39105 


96681 


53 


8 


55695 


58719 


41 281 


96976 


57607 


60931 


39069 


96676 


52 


9 


55728 


58757 


41243 


96971 


57638 


60967 


39033 


96670 


5J 


10 


1.55761 


1-58794 


0.41 206 


T.96 966 


1.57669 


T.61 004 


0.38 996 


T.96 665 


50 


II 


55 793 


58832 


41 168 


96962 


57700 


61 040 


38960 


96660 


49 


12 


"f^^ 


58869 


41 131 


96957 


57731 


61 076 


38924 


96655 


48 


13 


55858 


58907 


41093 


96952 


57762 


61 112 


38888 


96650 


47 


14 


55891 


58944 


41 056 


_ 96 947 


57 793 


61 148 


38852 


96645 


46 


'5 


1-55923 


1.58 981 


0.41 019 


1 .96 942 


1.57824 


T.61 184 


0.38816 


1.96640 


45 


i6 


55956 


59019 


40981 


96937 


57855 


61 220 


38780 


96634 


44 


17 


55988 


59056 


40944 


96932 


57885 


61 256 


38744 


96629 


43 


18 


56021 


59094 


40906 


96927 


57916 


61 292 


38708 


96 624 


42 


19 


56053 


59 131 


40869 


96922 


57 947 


61328 


38672 


96619 


41 


20 


1.56085 


1.59 168 


0.40 832 


T.96 917 


1-57978 


T.61 364 


0.38 636 


T.96 614 


40 


21 


56 118 


59205 


40795 


96912 


58008 


61 400 


38600 


96608 


39 


22 


56150 


59243 


40757 


96907 


58039 


61436 


38564 


96603 


38 


23 


56 182 


59 280 


40720 


96903 


58070 


61472 


38528 


96598 


37 


24 


56215 


59317 


40683 


96898 


58101 


61508 


38492 


96593 


36 


^1 


1.56247 


1-59 354 


0.40 646 


1.96893 


T.58 131 


T.61 544 


0.38 456 


1.96588 


35 


26 


56279 


59391 


40609 


96888 


58162 


61579 


38421 


96582 


34 


27 


563" 


59429 


40571 


96883 


58192 


61 615 


38385 


96577 


33 


28 


56343 


59466 


40534 


96878 


58223 


61651 


38349 


96572 


32 


29 


56375 


59503 


40497 


96873 


58253 


61687 


38313 


96567 


31 


30 


1.56408 


1.59540 


0.40 460 


1.96868 


1.58284 


T.6i 722 


0.38 278 


T.96 562 


30 


31 


56440 


59 577 


40423 


96863 


58314 


61758 


38242 


96556 


29 


32 


56472 


59614 


40386 


96858 


58345 


61794 


38206 


96551 


28 


33 


56504 


59651 


40349 


96853 


58375 


61830 


38170 


96546 


27 


■ 34 


56536 


59688 


40312 


96848 


58406 


61865 


38135 


96541 


26 


35 


T.56 568 


1-59725 


0.40 275 


T.96 843 


T.58 436 


1.61 901 


0.38099 


1.96535 


25 


36 


56599 


59762 


40238 


96838 


58467 


61 936 


38064 


96530 


24 


37 


56631 


59 799 


40 201 


96833 


58497 


61972 


38028 


96525 


23 


38 


56663 


59835 


40 165 


96828 


58527 


62008 


37992 


96520 


22 


39 


56695 


59872 


40 128 


96823 


58557 


62043 


37 957 


96514 


21 


40 


1.56727 


1.59 909 


0.40091 


T.96 818 


1.58588 


T.62 079 


0.37 921 


T.96 509 


20 


41 


56759 


59946 


40054 


96813 


58618 


62114 


37886 


96504 


19 


42 


56790 


59983 


40017 


96808 


58648 


62 150 


37850 


96498 


18 


43 


56822 


60019 


39981 


96803 


58678 


62 185 


37815 


96493 


17 


44 


56854 


60056 


39 944 


96798 


58709 


62 221 


37 779 


96488 


16 


45 


T.56 886 


T.60 093 


0.39 907 


1.96793 


1-58739 


T.62 256 


0-37 744 


T.96 483 


'5 


46 


56917 


60 130 


39870 


96788 


58769 


62 292 


37708 


96477 


14 


47 


56949 


60 166 


39834 


96783 


58799 


62327 


37673 


96472 


'3 


48 


56980 


60203 


39 797 


96778 


58 829 


62 362 


37638 


96467 


12 


49 


57012 


60240 


39760 


96772 


58859 


62398 


37602 


96461 


II 


50 


1.57044 


1.60 276 


0.39 724 


1.96767 


T.58 889 


1.62433 


0.37 567 


T.96 456 


10 


51 


57075 


60313 


39687 


96 762 


58919 


62468 


37532 


96451 


9 


52 


57107 


60349 


39651 


96757 


58949 


62504 


37496 


96445 


8 


53 


57138 


60386 


39614 


96752 


58979 


62539 


37461 


96440 


7 


54 


57169 


60422 


39578 


96747 


59009 


62574 


37426 


96435 


6 


55 


1.57 201 


T.60 459 


0.39 54« 


T.96 742 


1.59039 


1.62 609 


0.37 391 


1.96429 


5 


56 


57232 


60495 


39505 


96737 


59069 


62645 


37 355 


96424 


4 


57 


57264 


60532 


39468 


96732 


59098 


62680 


37320 


96419 


3 


58 


57295 


60568 


39432 


96727 


59128 


62715 


37285 


96413 


2 


59 


57326 


60605 


39 395 


96 722 


59158 


62 750 


37250 


96408 


I 


60' 


1-57358 


1.60 641 


0.39 359 


1.96717 


1.59188 


1.62 785 


0.37215 


1 .96 403 


0' 




log 00s 


log oot 


log tan 


log sin 


log cos 


log oot 


log tan 


log sin 


t 



LOG SIN, etc. 68<= 

66°-68° 



(58) 



67^ 



23= 



21°-24° 
24° LOG SIN, etc 



log sin log tan log cot log ops 



1-59 188 
59 2i» 

59247 
59277 

_ 59 307 

1-59336 

59366 

59396 

59425 

_ 59 455 

T.59 484 

59514 

59 543 

59 573 

59602 

7.59 632 
59661 
59690 

59720 
59 749 

1-59 778 
59808 

59837 
59866 

_ 59 895 

1-59924 

59 954 
59983 
60012 
60041 

Y.60 070 
60099 
60128 

60 157 
60186 

7.60 215 
60244 
60273 
60302 
60331 

T.60 359 
60388 
60417 
60446 

_ 60 474 

1.60503 
60532 
60561 
60589 
60618 

7.60 646 
60675 
60704 
60732 
60761 

T.60 789 
60818 
60846 
60875 

_ 60 903 

1.60931 



1.62 785 
62820 
6285s 
62890 
62926 

T.62 961 
62996 
63031 
63066 
63101 

7-63 13s 
63170 
63 205 
63240 
_ 63 275 
1-63310 
63345 
63379 
63414 
63449 

7.63 484 
63519 
63553 
63588 
63623 

7.63 657 
63692 

63 726 
63761 
63796 

7.63 830 
63865 
63899 
63934 

_ 63 968 
1.64003 

64037. 

64072 

64 106 
64 140 

7.6417s 
64209 

64243 

64278 

_ 64 312 

1 .64 346 
64381 
64415 
64449 

_ 64 483 
1.64517 
64552 
64586 
64620 
64654 
1 .64 688 
64 722 

64756 

64790 

64824 

7.64 858 



0.37 215 
37180 

3714s 
37 no 

37074 

0.37 039 

37004 

36969 

36934 
-36 899 

0.36 865 
36830 

36795 
36 760 

36725 
0.36 690 
36655 
36621 
36586 
36551 
0.36516 
36481 

36447 
36412 

36377 
0.36 343 
36308 
36274 
36239 
36204 

0.36 170 

36135 
36101 
36066 
36032 

0-35 997 
35963 
35928 
35894 
35860 

0.35 825 
35791 

■ 35 757 
35722 
35688 

0.35 654 
35619 
35585 
35551 
35517 

0.35 483 
35448 
35414 
35380 
35346 

0.35 312 
35278 
35244 
35210 

35 176 
0.3s '42 



1.96403 

96397 
96392 

96387 
_ 96 381 
1.96376 

96370 

96365 
96360 

96354 

7.96 349 

96343 

96338 

96333 

_ 96 327 

1.96322 

96316 

96311 

96305 

96300 

7.96 294 
96289 
96284 
96278 
96273 

1.96 267 
96262 
96256 
96251 
9624s 

T.96 240 
96234 
96229 
96223 
96218 

7.96212 
96207 
96 201 
96 196 
96 190 

7.96 185 
96179 
96174 
96168 
96 162 

1.96 157 
96151 
96 146 
96 140 

_ 96 135 

1.96 129 
96 123 
96118 
96 112 
96 107 

7.96 lOI 
96095 
96090 
96084 

_ 96 079 

1.96073 



log sis log tan log oot log cos 



1.60 931 
60960 
60988 
61 016 

_ 61 045 ' 

1. 61 073 
61 loi 
6i 129 
61 158 
61 186 

7.61 214 
61 242 
61 270 
61298 
61 326 

T-6i 354 
61382 
61 411 

61438 
61 466 

7.61 494 
61 522. 

61550 
61578 
61 606 

7.61 634 
61 662 
61689 
61 717 
61745 

7.61 773 
61 800 
61828 
61856 
61883 

7.61 911 

61939 

61 966 
61994 
62021 

7.62 049 
62076 

62 104 
62 131 
62 159 

7.62 186 

62 214 

62241 

62268 

62 296 

7.62 323 

62 350 

62377 

62405 

_ 62 432 

1.62459 

62486 

62513 

62 541 

_ 62 568 

1-62595 



1.64 858 
64892 

64 926 
64960 

•_ 64 994 

1.65 028 

65 062 
65 096 
65130 
65 164 

7.6s 197 
65231 
65 265 
65299 

_ 65 333 

1.65 366 
65 400 
65434 
65467 

_ 65 501 

165535 
65568 

65 602 
65636 

_ 65 669 

1.65 703 

65736 

65770 

65803 

_ 65 837 

1.65 870 
65904 
65937 
65971 
66004 

7.66 038 
66071 

66 104 
66138 
66 171 

7.66 204 
66238 
66 271 
66304 
_ 66 337 
1.66371 
66404 

66437 

66470 

_ 66 503 

1.66537 

66 570 

66603 

66636 

66669 

7.66 702 

66735 
66768 
66801 
66834 
7.66 867 



0.35 142 
35108 
35074 
35040 
35006 

0.34 972 
34938 
34904 
34870 
34836 

0.34 803 
34769 
34 735 
34701 
34667 

0.34 634 
34600 

34566 
34 533 
34499 

0.34 465 
34432 
34398 
34364 
34331 

0.34 297 
34264 
34230 
34197 
34163 

0.34 130 
34096 
34063 
34029 
33996 

0.33 962 
33929 
33896 
33862 
33829 



1.96073 
96067 
96062 
96056 
96050 

7.96 045 
96039 
96034 
96028 
96022 

7.96017 
96 on 
96005 
96000 
95 994 

7.95 988 
95982 
95 977 
95971 
95965 

T-95 960 
95 954 
95948 
95942 
_ 95 937 
1-95931 
95925 
95920 

95914 
95908 

7.95 902 

95897 
95891 

95885 
_ 95 879 
1-95873 
95868 
95 862 
95856 
95850 



0.33 796 


1-95844 


33762 


95839 


33729 


95833 


33696 


95827 


33663 


95821 


0.33 629 


1-95 815 


33596 


95 810 


33563 


95804 


33530 


• 95 798 


33 497 


95792 


0.33463 


1.95786 


33430 


95780 


33 397 


95 775 


33364 


95769 


33331 


95763 


0.33 298 


1-95 757 


33265 


95751 


33232 


95 745 


33199 


95 739 


33166 


95 733 


0.33 133 


1.95728 



log COS 



log cot log tan log sin | log cos log cot log tan log sin 



60' 

59 

58 
57 
56 
55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 

41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
3' 

30 

29 

28 

27 
26 

25 
24 
23 

22 
21 

20 

19 
18 

17 
16 

«S 
14 
13 
12 
11 

10 

9 

8 

7 
6 

5 
4 
3 
2 



66° 



(59) 



Afio LOG SIN, etc. 



25°-28^ 
LOG SIN, etc. 26*^ 



26^ 



t 


log sin 


log tan 


log cot 


log cos 


log sin 


log tan 


log oot 


log cos 


^^^" 


0' 


T.62 595 


r.66 867 


0-33 133 


7.95 728 


T.64 184 


7.68818 


0.31 182 


7-95 366 


60' 


I 


62622 


66900 


33100 


95722 


64210 


68850 


31 150 


95 360 


59 


2 


62649 


66933 


33067 


95716 


64 236 


68882 


31 118 


95 354 


- -58- 


3 


62676 


66966 


33034 


95710 


64262 


68914 


31086 


95348 


57 


4 


62703 


66999 


33 001- 


95704 


64288 


68946 


3' 054 


95341 


56 


5 


1.62730 


T.67 032 


0.32 968 


1.95 698 


7.64313 


7.68 978 


0.31 022 


1-95 335 


55 


6 


62757 


67065 


32935 


95692 


64339 


69010 


30990 


95329 


54 


7 


62784 


67098 


32902 


95686 


64365 


69042 


30958 


95323 


53 


8 


62 8u 


67 131 


32869 


95680 


64391 


69074 


30926 


95317 


52 


9 


62838 


67163 


32837 


95674 


64417 


69 ie6 


30894 


95310 


51 


10 


r.62 865 


1.67 196 


0.32 804 


7.95 668 


T.64 442 


7.69 138 


0.30 862 


7.95 304 


50 


n 


62892 


67229 


32771 


95663 


64468 


69 170 


30830 


95298 


49 


12 


62918 


67 262 


32738 


95657 


64494 


69 202 


30798 


95292 


48 


'3 


62945 


67295 


32705 


95651 


64519 


69234 


30766 


95286 


47 


14 


62972 


67327 


32 673 


95645 


64545 


69266 


30734 


95279 


46 


IS 


Y.62 999 


1.67360 


0.32 640 


1.95639 


1.64 571 


1.69 298 


0.30 702 


1-95273 


45 


i6 


63026 


67393 


32607 


95633 


64596 


69329 


30671 


95267 


44 


17 


63052 


67426 


32574 


95627 


64622 


69361 


30639 


95261 


43 


18 


63079 


67458 


32542 


95621 


64647 


69393 


30607 


95254 


42 


19 


63 1 06 


67491 


32509 


95615 


64673 


69425 


3057s 


95248 


41 


20 


7.63 133 


T.67 524 


0.32476 


1.95609 


T.64 698 


7.69457 


0-30 543 


7.95 242 


40 


21 


63159 


67556 


32444 


95603 


64724 


69488 


30512 


95236 


39 


22 


63186 


67589 


32 411 


95 597 


64749 


69 520 


30480 


95229 


38 


23 


63213 


67 622 


32378 


95591 


64775 


69552 


30448 


95223 


37 


24 


63239 


67654 


32346 


95585 


64800 


69584 


30416 


95217 


36 


25 


1.63 266 


1.67687 


0.32313 


1-95 579 


T.64 826 


1.69 615 


0.30 385 


1.95211 


35 


26 


63292 


67719 


32281 


95 573 


64851 


69647 


30353 


95204 


34 


27 


63319 


67752 


32248 


95567 


64877 


69679 


30321 


95198 


33 


28 


63345 


67 785 . 


32215 


95561 


64902 


69 710 


30290 


95192 


32 


29 


63372 


67817 


32183 


95 555 


64927 


69742 


30258 


95185 


31 


30 


1-63398 


1.67850 


0.32 150 


1-95 549 


7.64 953 


T.69 774 


0.30 226 


1-95 179 


30 


31 


63425 


67882 


32 118 


95 543 


64978 


69805 


30195 


95173 


29 


32 


63451 


67915 


32085 


95 537 


65003 


69837 


30163 


95167 


28 


2i 


63478 


67947 


32053 


95531 


65029 


69868 


30132 


95 160 


27 


34 


63504 


67980 


32020 


95525 


65054 


69900 


30100 


95154 


26 . 


35 


1-63531 


T.68012 


0.31 988 


1-95 5'9 


1.65079 


7.69 932 


0.30 068 


1.95 148 


25 


36 


63557 


68044 


31956 


95513 


65 104 


69963 


30037 


95141 


24 


37 


63583 


68077 


31923 


95507 


65130 


69995 


30005 


95135 


23 


38 


63 610 


68109 


31 891 


95500 


65155 


70026 


29974 


95129 


22 


39 


63636 


68142 


31858 


95 494 


65 180 


70058 


29942 


95 122 


21 


40 


T.63 662 


T.68174 


0.31 826 . 


7.95 488 


7.65 205 


7.70 089 


0.29911 


7.95 116 


20 


41 


63689 


68206 


31 794 


95482 


65 230 


70 121 


29879 


-95 no 


19 


42 


63715 


68239 


31761 


95476 


6525s 


70152 


29848 


95 103 


18 


43 


63741 


68271 


31729 


95470 


65281 


70 184 


29816 


95097 


17 


44 


63767 


68303 


31697 


95464 


65306 


70215 


29785 


95090 


16 


45 


1-63794 


7.68 336 


0.31 664 


1-95458 


1-65331 


1.70247 


0.29 753 


1.95084 


'5 


46 


63820 


68368 


3' 632 


95452 


65 356 


70278 


29722 


95078 


«4 


47 


63846 


68400 


31 600 


95446 


65381 


70309 


29691 


95071 


13 


48 


63873. 


68432 


31568 


95440 


65 406 


70341 


29659 


95065 


12 


49 


63898 


68465 


31535 


95 434 


65431 


70372 


29628 


95059 


11 


50 


T.63 924 


T.68 497 


0-31 503 


1-95427 


1.65456 


1.70404 


0.29 596 


7.95 052 


10 


51 


63950 


68529 


3' 47' 


95421 


65481 


70435 


29565 


95046 


9 


52 


63976 


68561 


31439 


95415 


65 506 


70466 


29534 


95039 


8 


S3 


64002 


68593 


3t4<?7 


95409 


65531 


70498 


29502 


95033 


7 


54 


64028 


68626 


3«374 


95403 


65556 


70529 


29471 


95027 


6 


55 


T.64 054 


T.68 658 


0.31 342 


1-95 397 


1.65 580 


1.70560 


0.29440 


1.95020 


5 


56 


64080 


68690 


3I3JO 


95 391 


65605 


70592 


29408 


95014 


4 


57 


64 106 


68722 


31278 


95384 


65630 


70623 


29377 


95007 


3 


S8 


64132 


68754 


31246 


95378 


^5^55 


70654 


29346 


95001 


2 


59 


64158 


68786 


31 214 


95372 


65680 


70685 


29315 


94 995 


1 


60' 


1.64 184 


7.68818 


0.31 182 


1.95366 


1.65 70s 


1.70 717 


0.29 283 


1.94988 


0' 




log COB 


log oot 


log tan 


log sin , 


log cos 


log oot 


log tan 


log sin 


1 



LOG SIN, etc. 64° 
61°-64° 



(60) 



63= 



27° 



0' 

I 

2 

3 
4 

S 
6 

7 
8 

9 
10 



13 
H 

«S 
i6 

'7 
i8 

19 

20 

21 

22 

23 
24 

25 
26 

27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 

41 
42 

43 
44 

45 
46 

47 
48 

49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60' 



log sin log tan logoot log cos I log sin 



26°-28° 
28° LOG SIN, etc 



1.65 705 
65 729 
65754 
65779 
65 804 

T.65 828 

65853 
65878 

65 902 
65927 

if -65 952 
65976 
66001 
66025 
66050 

T.66 075 
66099 

66 124 
66148 
66173 

T.66 197 

66221 

66246 

66270 

_ 66 295 

1.66319 

66343 
66368 
66392 
66416 

1.66 441 
66465 
66489 
66513 

_ 66 537 
1.66562 
66586 
66610 
66634 
66658 

T.66 682 
66 706 
66 731 
66755 

66 779 
T.66 803 

66827 
66851 
66875 
$6899 

T.66 922 
66946 
66970 
66994 
67018 

T.67 042 
67066 
67090 

67 113 

_ 67 137 

1.67 161 



log cos 



1.70 717 

70748 
70779 

70810 
70841 

T.70 873 
70904 
70935 

70966 

70997 

T.7I 028 

71059 

71 090 

71 T2I 

_7ii53 

1. 71 184 

71 215 

71 246 

71277 

_ 71 308 

1-71339 
71370 
71401 

71431 
71 462 

7.71 493 
7«524 
71555 
71586 

71 617 

T.71 648 

71679 

71709 

71740 

_ 71 771 

1. 71 802 

71833 
71863 
71894 
71925 

I-7I 955 
71986 
72017 
72048 
72078 

T.72 109 

72 140 
72170 
72201 
72231 

T.72 262 
72293 
72323 
72354 
_ 72 384 
1-72415 
72445 
72476 
72 506 

_ 72 537 

1-72567 

log cot 



0.29 283 
29 252 

29 221 
29 190 
29159 
0.29 127 
29096 
29065 
29034 
29003 

0.28 972 
28941 
28910 
28879 
28847 

0.28816 
28785 
28754 
28 723 
28 692 

0.28 661 
28 630 

28599 
28569 
28538 
0.28 507 
28476 

28445 
28414 
28383 

0.28 352 
28 321 
28291 
28 260 
28229 

0.28 198 
28167 
28137 
28106 
28075 

0.28 045 
28014 

27983 
27952 
27922 
0.27 891 
27860 
27 830 
27799 
27 769 

0.27 738 

27707 
27677 
27 646 
27 616 
0.27 585 

27555 
27524 

27494 

27463 

027433 



log tan 



1 .94 988 
94982 

94 975 
94969 
94962 

1.94956 
94 949 
94 943 
94936 
94930 

T.94 923 
94917 

94 9" 
94904 
94898 
T.94 891 
94885 
94878 
94871 
94865 

T.94 858 
94852 
94845 
94839 
_ 94 832 
1.94826 
94819 

94813 
94806 

94 799 

7.94 793 

94786 

94780 

94 773 

_ 94 767 

1.94760 

94 753 

94 747 

94740 

94 734 

T.94 727 

94720 

94714 

94707 

_ 94 700 

1.94694 

94687 

94680 

94674 

94667 

T.94 660 
94654 
94647 
94640 

_ 94 634 

1.94627 
94620 
94614 
94607 
94600 

T.94 593 
log sin 



log tan log cot log ooa 



1.67 161 

67 185 

67 208 

67232 

_ 67 256 

1.67 280 

67303 

67327 

67350 

_ 67 374 

T.67 398 

67 421 

67445 
67468 

_ 67 492 

1-67515 
67539 
67562 
67586 
67 609 

i"-67 633 
67 656 
67680 
67703 
67 726 

T.67 750 

67773 
67 796 
67 820 
67843 
T.67 866 

67 890 

67913 
67936 

_ 67 959 
1.67982 
68006 
68029 
68052 
68075 

T.68 098 

68 121 
68 144 
68167 
68 190 

T.68 213 
68237 
68260 
68283 
68305 

T.68 328 
68351 
68374 
68397 
68420 

T.68 443 
• 68466 
68489 
68512 
68534 
1-68557 
log cos 



1.72567 
72598 
72628 
72659 
72 689 

T.72 720 

72750 

72 780 
72 811 

72 841 

T.72 872 
72902 

72932 
72963 

_ 72 993 
1.73023 

73054 
73084 

73 "4 
73144 

1-73175 
73205 

73235 
73265 

_ 73 295 

1-73326 

73356 

73386 

73416 

73446 

T.73 476 

73507 

73 537 

73567 

_ 73 597 

1.73627 

73657 
73687 

73717 
_ 73 747 

1-73 777 
73807 

73837 
73867 

_ 73 897 
1.73927 

73 957 
73987 
74017 
74047 

1-74077 
74107 

74137 

74 166 

_ 74 196 

1.74226 

74256 

74286 

74316 

_ 74 345 

1-74 375 



0-27 433 
27402 
27372 

27341 
27 311 

0.27 280 
27 250 
27 220 
27 189 
27159 

0.27 128 
27098 
27068 
27037 
27007 

0.26977 
26946 
26916 
26 886 
26856 

0.26 825 

26795 
26 765 

26735 
26705 

0.26 674 
26644 
26614 
26584 
26554 

0.26 524 

26493 
26463 

26433 
26403 

0.26 373 

26343 
26313 
26283 
26253 

0.26 223 
26193 
26 163 
26 133 
26 103 

0.26 073 
26 043 
26013 
25983 
25953 

0.25 923 

25893 
25863 

25834 
25 804 

0.25 774 

25744 
25714 
25 684 

2565s 
0.25 625 



1-94 593 
94587 
94580 

94 573 
_ 94 567 
1.94560 

94 553 

94546 

94540 

_ 94 533 

1.94526 

94519 

94513 

94506 

_ 94 499 

1.94492 

.94485 
94479 

94472 

_ 94 465 

T.94 458 

94451 

94 445 

94438 

_ 94 431 

1.94424 

94417 
94410 

94404 

94 397 

T.94 390 
94383 
94376 
94369 

_ 94 362 

1-94 355 
94 349 
94342 
94 335 
94328 

T.94 321 
94314 
94307 
94300 

_ 94 293 
1 .94 286 
94279 
94273 
94266 
94259 
T.94 252 

94245 

94238 

94231 

94224 

1.94217 

94 210 

94203 

94196 

_94i89 

1.94 182 



log cot log tan log sin 



62^^ 



(61) 



AI0 LOG SIN, etc. 
• 6r-64° 



29°-32° 
LOG SIN, etc. »9 



30° 



/ 


log sin 


log tan 


log oot 


log oos 


log sin 


log tan 


log oot 


log 008 




0' 


'•^f557 


1-74 375 


0.25 625 


r.94 182 


7.69 897 


7.76 144 


0.23 856 


T.93 753 


60' 


I 


68580 


74405 


25595 


94175 


69919 


76173 


23827 


93746 


59 


2 


68603 


74 435 


25565 


94168 


69941 


76202 


23798 


93738 


58 


3 


68625 


74465 


25 535 


94 161 


69963 


76231 


23769 


93731 


57 


4 


68648 


74 494 


25 506 


94154 


69984 


76261 


23739 


93724 


56 


S 


7.68671 


1.74524 


0.25 476 


1.94147 


7.70 006 


7.76 290 


0.23 710 


1-93717 


55 


6 


68694 


74 554 


25446 


94140 


70028 


76319 


23681 


93709 


54 


7 


68 716 


74583 


25417 


94133 


70050 


76348 


23652 


93702 


53 


8 


68739 


74613 


25387 


94126 


70072 


76377 


23623 


93695 


52 


9 


68762 


74643 


25357 


94119 


70093 


76406 


23594 


93687 


51 


10 


T.68 784 


^■74673 


0.25 327 


T.94 112 


7.70 115 


7.76435 


0,23 565 


1.93680 


50 


II 


68807 


74702 


25 298 


94105 


70137 


76464 


23536 


93673 


49 


12 


68829 


74732 


25268 


94098 


70159 


76493 


23507 


93665 


48 


13 


68852 


74762 


25238 


94090 


70 180 


76522 


23478 


93658 


47 


14 


68875 


74791 


25209 


94083 


70202 


76551 


23449 


93650 


46 


'5 


1.68897 


1.74 821 


0.25 179 


1.94076 


7.70 224 


1.76580 


0.23 420 


1-93643 


45 


i6 


68920 


74851 


25149 


94069 


70245 


76609 


23391 


93636 


44 


17 


68942 


74880 


25 120 


94062 


70267 


76639 


23361 


93628 


43 


18 


68965 


74910 


25090 


9405s 


70288 


76668 


23332 


93621 


42 


19 


68987 


74 939 


25061 


94048 


70310 


76697 


23303 


93614 


41 


20 


T.69010 


1.74969 


0.25 031 


T.94 041 


7.70 332 


1.76725 


0.23 275 


7.93 606 


40 


21 


69032 


74998 


25002 


94034 


70353 


76754 


23246 


93 599 


39 


22 


69055 


75028 


24972 


94027 


70375 


76783 


23217 


93 591 


38 


23 


69077 


75058 


24942 


94020 


70396 


76812 


23188 


93584 


37 


24 


69 100 


75087 


24913 


94012 


70418 


76841 


23159 


93 577 


36 


^1 


T.69 122 


1.75 117 


0.24 883 


7.94 005 


7.70439 


7.76 870 


0.23 130 


1-93569 


35 


26 


69144 


75146 


24854 


93998 


70461 


76899 


23IOI 


93562 


34 


27 


69 167 


75176 


24824 


93991 


70482 


76928 


23072 


93 554 


33 


28 


69 189 


75205 


24795 


93984 


70504 


76957 


23043 


93 547 


32 


29 


69212 


75235 


24765 


93 977 


70525 


76986 


23014 


93 539 


31 


30 


T.69 234 


1.75264 


0.24 736 


1.93970 


1-70547 


7.77015 


0.22 985 


1-93532 


30 


31 


69 256 


75294 


24 706 


93963 


70568 


77044 


22956 


93525 


29 


32 


69279 


75323 


24677 


93 955 


70590 


77073 


22927 


93517 


28 


33 


69301 


75 353 


24647 


93948 


70 61 1 


77101 


22899 


93510 


27 


34 


69323 


75382 


24618 


93941 


70633 


77130 


22870 


93502 


26 


35 


1.69345 


1.75411 


0.24 589 


■•93 934 


1.70654 


1.77159 


0.22 841 


1-93 495 


25 


36 


69368 


75441 


24559 


93927 


70675 


77188 


22812 


93487 


24 


37 


69390 


75470 


24530 


93920 


70697 


77217 


22783 


93480 


23 


38 


69412 


75500 


24500 


93912 


70718 


77246 


22754 


93472 


22 


39 


69434 


75529 


24471 


93905 


70739 


77274 


22 726 


93465 


21 


40 


T.69 456 


1-75558 


0.24 i\i\?. 


7.93 898 


7.70 761 


1-77303 


0.22 697 


7-93 457 


20 


41 


69479 


75588 


24412 


93891 


70 782 


77332 


22668 


93450 


'R 


42 


69 501 


75617 


24383 


93884 


70803 


77361 


22639 


93442 


18 


43 


69523 


75647 


24353 


93876 


70824 


77390 


22610 


93435 


17 


44 


69545 


75676 


24324 


93869 


70846 


77418 


22582 


93427 


16 


45 


1.69567 


1-75 70s 


0.24 295 


1.93 862 


7.70 867 


1-77 447 


0.22 553 


1.93420 


15 


46 


69589 


75 735 


24265 


93855 


70888 


77476 


22524 


93412 


14 


47 


69 611 


75764 


24236 


93847 


70909 


77505 


22495 


93405 


13 


48 


69633 


75 793 


24207 


93840 


70931 


77 533 


22467 


93 397 


12 


49 


69655 


75822 


24178 


93833 


70952 


77562 


22438 


93390 


II 


50 


1.69677 


1-75852 


0.24 148 


1.93826 


1-70973 


1-77 591 


0.22 409 


1.93382 


10 


SI 


69699 


75881 


24119 


93819 


70994 


77619 


22381 


93 375 


9 


52 


69721 


75910 


24090 


93 811 


71 015 


77648 


22352 


93367 


8 


53 


69743 


75 939 


24061 


93804 


71036 


77677 


22323 


93360 


7 


54 


69765 


75969 


24031 


93 797 


71058 


77706 


22294 


93352 


6 


55 


7.69 787 


1.75998 


0.24 002 


1-93789 


1.71 079 


1-77 734 


0.22 266 


1-93 344 


5 


56 


69809 


76027 


23973 


93782 


71 100 


77763 


22 237 


93 337 


4 


57 


69831 


76056 


23944 


93 775 


71 121 


77791 


22 209 


93329 


3 


58 


69853 


76086 


23914 


93768 


71 142 


77 820 


22 180 


93322 


2 


59 


69875 


76 115 


23885 


93760 


71 163 


77849 


22 151 


93314 


1 


60' 


1.69897 


1.76144 


0.23 856 


1-93 753 


1. 71 184 


1-77877 


0.22 123 


1-93307 


0' 




log cos 


log oot 


log tan 


log sin 


log oos 


log oot 


log tan 


log sin 


t 



LOG SIN, etc. 60*^ 

67°-60° 



(62) 



59° 



t 


31° 

loff sin loff tan ^no• nn¥ Tftm »»« 


32° log"in!^ 




9tC 


0' 

I 

2 

3 

4 


ea *" iug uoiu iO^ oOii ■'Off COS 

1.71184 1.77 S77 0.22123 I.93 307 
71205 77906 22094 93299 
71226 77935 22065 93291 
71247 77963 22037 93284 

_ 71 268 _ 77 992 22008 93276 


log sin log tan log cot log cos 
1.72421 1.79579 0.20421 T.92842 
72 441 79 607 20 393 92 834 
72461 79635 20365 92826 
72482 79663 20337 92818 
72502 79691 20309 92810 


60' 

59 
58 
57 
56 

55 
54 
SZ 
52 
51 




I 


1. 71 289 1.78020 0.21980 T.93 269 
71 310 78049 21 951 93261 
71 331 78077 21923 93253 
71 352 78 106 21 894 93 246 


1.72522 1.79719 0.20281 7.92803 
72542 79747 20253 92795 
72562 79776 20224 92787 
72 582 79 804 20 196 92 779 
72 602 79 832 20 168 92 771 




9 


71373 7813s 21865 93238 




10 


1-71393 1-78 163 0.21837 1.93230 


T.72 622 r.79 860 0.20 140 r.92 763 


50 




II 


71 414 78 192 21 808 93 223 


72643 79888 20112 92755 


49 




12 


71 435 78 220 21 780 93 215 


72663 79916 20084 92747 


48 




'3 


71456 78249 21 751 93207 


72683 79944 20056 92739 


. 47 




«4 


71477 78277 21723 93200 


_ 72 7°3 79 972 20 028 92 731 


46 




'5 


1.71498 1.78306 0.21694 1-93192 


1.72723 1.80000 0.20000 r.92 723 


45 




l6 


71 519 78334 21666 93184 


72743 80028 19972 92715 


44 




•'2 


71539 78363 21637 93177 


72763 80056 19944 92707 


43 




i8 


71 560 78 391 21 609 93 169 


72783 80084 19916 92699 


42 




19 


71581 78419 21 581 93 161 


72 803 80 1 12 19 888 92 691 


41 




20 


1. 71 602 T. 78 448 0.21552 T.93 154 


1.72823 r.80140 0.19860 T.92683 


40 




21 


71622 78476 21524 93146 


72 843 80 168 19 832 92 675 


39 




22 


71643 78505 21495 93138 


72 863 80 195 19 805 92 667 


38 




23 


71664 78533 21467 93 131 


72883 80223 19777 92659 


37 




24 


71685 78562 21438 93123 


72902 80251 19749 92651 


36 




25 


1.71705 1.78590 0.21 410 1.93 115 


1.72922 1.80279 0.19 721 r.92 643 


35 




26 


71 726 78 618 21 382 g^ 108 


72942 80307 19693 92635 


34 




27 


71 747 78 647 21 353 93 100 


72962 80335 19665 92627 


33 




28 


71767 78675 21325 93092 


72982 80363 19637 92619 


32 




29 


71 788 78 704 21 296 93 084 


73002 80391 19609 92611 


31 




30 


T.71809 1.78732 0.21268 r.93077 


1.73022 T.80419 0.19 581 T.92603 


30 




31 


71829 78760 21240 93069 


73041 80447 19553 92595 


29 




32 


71850 78789 21 211 93061 


73061 80474 19526 92587 


28 




33 


71870 78817 21 183 93053 


73081 80502 19498 92579 


27 




34 


71 891 78845 21 155 93046 


73101 80530 19470 92571 


26 




^i 


1.71911 1.78874 0.21 126 1.93038 


T. 73 121 1.80558 0.19442 1.92563 


25 




36 


71932 78902 21098 93030 


73140 80586 19414 92555 


24 




37 


71952 78930 21070 93022 


73 160 80 614 19 386 92 546 


23 




38 


7' 973 78959 21 041 93014 


73180 80642 19358 92538 


22 




39 


71994 78987 21 013 93007 


73200 80669 19331 92530 


21 




40 


T.72014 T. 79 015 0.20985 T.92999 


T. 73 219 T.80697 0.19303 T.92522 


20 




41 


72034 79043 20957 92991 


73239 80725 19275 92514 


19 




42 


7205s 79072 20928 92983 


73259 80753 19247 92506 


18 




43 


72075 79100 20900 92976 


73278 80781 19 219 92498 


17 




44 


72096 79128 20872 92968 


73 298 80 808 19 192 92 490 


16 




1^ 


1.72 116 1.79 156 0.20844 1.92960 


1.73318 1.80836 0.19 164 T.92482 


15 




46 


72137 79185 20815 92952 


73 337 80864 19 136 92473 


14 




47 
48 


72157 79213 20787 92944 


73357 80892 19 108 92465 


13 




72177 79241 20759 92936 


73 377 80919 19081 92457 


12 




49 


_ 72 198 79269 20731 "92929 


73396 80947 19053 92449 


II 




50 


1.72 218 T.79297 0.20703 T.92921 


1.73 416 1.80975 0.19025 1.92441 


10 




SI 


72238 79326 20674 92913 


73435 81003 18997 92433 


9 




52 


72259 79 354 20646 92905 


73455 81030 18970 92425 


8 




S3 


72279 79382 20618 92897 


73474 81058 18942 92416 


7 




54 


72299 79 410 20590 92889 


73 494 81 086 18 914 92 408 


6 




55 


1.72320 1.79438 0.20562 T.92881 


T.73513 T.81 113 0.18887 1.92400 


s 




56 


72340 79466 20534 92874 


73533 81141 18859 92392 


4 




57 


72360 79495 20505 92866 


73 552 81 169 18 831 92 384 


3 




58 


72381 79523 20477 92858 


73572 81 196 18804 92376 


2 




kl, 


72401 79551 20449 92850 


73591 81224 18776 92367 


I 




60' 


1.72 421 1.79579 0.20421 1.92842 


1.73611 1.81252 0.18748 1.92359 


0' 






log cos log cot log tan log sin 


log 003 log oot log tan log sin 


t 






68° (6 


3) 570 LOG S 

0/ 


N, e 


tc. 



33°-36^ 
LOG SIN, etc. 


33° 








34° 








f 


log sin 


log tan 


log oot 


log cos 


log sin 


log tan 


log cot 


log 003 






0' 


1-73611 


1.81 252 


18748 


1-92359 


7-74 756 


7.82 899 


0.17 101 


7-91 857 


60' 




I 


73630 


81279 


18721 


92351 


74 775 


82926 


17074 


91849 


59 




2 


73650 


81307 


18693 


92343 


74 794 


82953 


17047 


91 840 


58 




3 


73669 


81335 


18665 


92335 


74812 


82980 


17020 


91832- 


57 




4 


73689 


81362 


18638 


92326 


74831 


83008 


16992 


91823 


56 




S 


1.73708 


T.81 390 


18610 


1.92318 


1.74850 


^-83035 


0.16 965 


1.91 815 


55 




6 


73727 


81 418 


18582 


92 310 


74868 


83062 


16938 


91 806 


54 




7 


73 747 


81445 


1855s 


92302 


74887 


83089 


I69I1 


91798 


53 




8 


73766 


81473 


18527 


92293 


74906 


83117 


16883 


91789 


52 




9 


73785 


81 500 


18500 


92285 


74924 


83144 


16856 


91 781 


51 




10 


1.73805 


T.8i 528 


18472 


7.92 277 


1-74 943 


7.83171 


0.16829 


7.91 772 


50 




II 


73824 


81556 


18444 


92 269 


74961 


83198 


16802 


91763 


49 




12 


73843 


81583 


18417 


92 260 


74980 


83225 


16775 


9175s 


48 




13 


"f^3 


81 611 


18389 


92 252 


74 999 


83252 


16748 


91746 


47 




H 


73882 


81638 


18362 


92244 


75017 


83280 


16 720 


91738 


46 




15 


1.73901 


T.81 666 


18334 


1.92235 


1.75036 


T-83 307 


0.16 693 


1-91729 


45 




16 


73921 


81693 


18307 


92 227 


75054 


83334 


16666 


91720 


44 




17 


73940 


81 721 


18279 


92219 


75073 


83361 


16639 


91712 


43 




18 


73 959 


81 748 


18 252 


92211 


75091 


83388 


16612 


91703 


42 




19 


73978 


81776 


18224 


92202 


75 no 


83415 


16585 


91695 


41 




20 


1-73 997 


1. 81 803 


18197 


7.92 194 


1.75 128 


7.83 442 


0.16558 


7.91 686 


40 




21 


74017 


81831 


18169 


92186 


75147 


83470 


16530 


91677 


39 




22 


74036 


81858 


18 142 


92177 


75 '65 


83497 


16503 


91 669 


38 




23 


74055 


81886 


18 114 


92 169 


75184 


83524 


16476 


91 660 


37 




24 


74074 


81913 


18087 


92 161 


75202 


83551 


16449 


91651 


36 




^5 


1-74093 


1. 81 941 


18059 


7.92 152 


1-75221 


1.83578 


0.16422 


1-91 643 


35 




26 


74 113 


81968 


18032 


92144 


75239 


83605 


16395 


91634 


34 




27 


74132 


81 996 


18004 


92136 


75258 


83632 


16368 


91625 


33 




28 


74 151 


82023 


17977 


92 127 


75276 


l^^M 


16341 


91617 


32 




29 


74170 


82051 


17949 


92119 


75294 


83686 


16314 


91608 


31 




30 


1.74 189 


T.82078 


17922 


1.92 III 


1-75313 


T.83 713 


0.16287 


7.91 599 


30 




31 


74208 


82106 


17894 


92 102 


75331 


83740 


16260 


91591 


29 




32 


74227 


82133 


17867 


92094 


75350 


83768 


16232 


91582 


28 




33 


74246 


82161 


17839 


92086 


75368 


8379s 


16205 


91573 


27 




34 


74265 


82188 


17812 


92077 


75386 


83822 


16 178 


91565 


26 




35 


1.74284 


1.82 215 


17785 


1.92069 


1-75405 


7.83 849 


0.16 151 


1-91556 


25 




36 


74303 


82243 


17757 


92060 


75423 


83876 


16 124 


91547 


24 




37 


74322 


82 270 


17730 


92052 


75441 


83903 


16097 


91538 


23 




38 


74341 


82298 


17702 


92044 


75 459 


83930 


16070 


91530 


22 




39 


74360 


82325 


1767s 


92035 


75478 


83957 


16043 


91 521 


21 




40 


1-74 379 


1.82352 


17648 


7.92 027 


1-75496 


1.83 984 


0.16016 


1.91 512 


20 




41 


74398 


82380 


17 620 


92018 


75514 


84011 


15989 


91504 


19 




42 


74417 


82407 


17593 


92010 


75 533 


84038 


15962 


91495 


18 




43 


74436 


82435 


17565 


92002 


75551 


84065 


15935 


91 486 


17 




44 


74 455 


82462 


17538 


91993 


75569 


84092 


15908 


91477 


16 




45 


1.74474 


7.82489 


17511 


1.91 985 


1-75 587 


7.84 119 


0.15881 


1.91 469 


15 




46 


74 493 


82517 


17483 


91976 


75605 


84 146 


15854 


91 460 


14 




47 


74512 


82544 


17456 


91 968 


75624 


84173 


15827 


91451 


13 




48 


74531 


82571 


17429 


91959 


75642 


84200 


15800 


91442 


12 




49 


74 549 


82599 


17401 


91951 


75660 


84227 


15773 


91433 


II 




50 


T-74 568 


7.82 626 


17374 


7.91 942 


1.75678 


7.84 254 


0.15 746 


7.91 425 


10 




51 


74587 


82653 


17347 


91934 


75696 


84280 


15720 


91 416 


9 




52 


74606 


82681 


17319 


91925 


75714 


84307 


15693 


91407 


8 




S3 


74625 


82708 


17292 


91917 


75 733 


84334 


15666 


91398 


7 




54 


74644 


8273s 


17265 


91 908 


75751 


84361 


15639 


91389 


6 




55 


1.74662 


1.82762 0. 


17238 


7.91 900 


1.75 769 


7.84 388 


0.15 612 


1.91381 


5 




56 


74681 


82790 


17 210 


91 891 


75787 


84415 


15585 


91372 


4 




57 


74700 


82817 


17183 


91883 


75805 


84442 


15558 


91363 


3 




58 


74719 


82844 


17156 


91874 


75823 


84469 


15531 


91354 


2 




59 


74 737 


82871 


17129 


91866 


75841 


84496 


15504 


91345 


I 




60' 


1-74756 


1.82899 0. 


17 101 


1-91 857 


1.75 859 


1.84523 


0-15477 


1.91336 


0' 






log cos 


log cot 


'og tan 


log sin 


log COS 


log OOt 


log tan 


log sin 


1 


LC 
i 


G SI 
J3°-5 


N, etc. 
6° 


66° 




(6 


4) 


66"" 











33 


°-36° 






35° 


36° LOG SIN, etc. 


t 


log sin log tan log cot log 00s 


log sin log tan log cot log cos 






0' 


1.75859 1.84523 0.15477 1.91336 


T.76 922 T.86 126 0.13874 T.90 796 


60' 




I 


75877 84550 15450 91328 


76939 86153 13847 90787 


59 




2 


75895 84576 15424 91319 


76957 86179 13 821 90777 


58 




3 


75913 84603 15397 91 310 


76 974 86 206 13 794 90 768 


57 




4 


75931 84630 15370 91 301 


76991 86232 13768 90759 


56 




s 


1.75949 1.84657 0.15343 1.91 292 


1.77009 1.86259 0.13741 1.90750 


55 




6 


75967 84684 15316 91283 


77026 86285 13715 90741 


54 




7 


75985 84 71 1 15289 91274 


77043 86312 13688 90731 


53 




8 


76003 84738 15262 91266 


77061 86338 13662 90722 


52 




9 


76021 84764 15236 91257 


77078 86365 13635 90713 


51 




10 


T.76039 T.84791 0.15209 1.9 1 248 


T.7709S T.86 392 0.13608 T.90 704 


50 




II 


76057 84818 15 182 91239 


77112 86418 13582 90694 


49 




12 


7607s 8484s 15155 91230 


77130 86445 13555 90685 


48 




13 


76093 84872 15128 91221 


77147 86471 13529 90676 


47 




14 


76 III 84899 IS 101 91 212 


77164 86498 13502 90667 


46 




IS 


T.76129 1.84925 0.15075 1.91203 


T.77i8i T.86 524 0.13476 1.90657 


45 




16 


76146 84952 15048 91 194 


77199 86551 13449 90648 


44 




17 


76164 84979 15021 91185 


77216 86577 13423 90639 


43 




18 


76182 85006 14994 91 176 


77233 86603 13397 90630 


42 




19 


76200 85033 14967 91 167 


77250 86630 13370 90620 


41 




20 


T.76218 T.85059 0.14 941 r.91 158 


T.77268 T.86 656 0.13344 T.90611 


40 




21 


76236 85086 14914 91149 


77285 86683 13 317 90602 


39 
38 




22 


76253 85113 14887 91141 


77302 86709 13 291 90592 




23 


76 271 85 140 14 860 91 132 


77319 86736 13264 90583 


^l 




24 


76 289 85 166 14 834 91 123 


77336 86762 13238 90574 


36 




25 


T.76307 1.85193 0.14807 1.91 114 


T.77353 1-86789 0.13211 1.90565 


35 




26 


76 324 85 220 14 780 91 105 


77370 86815 13185 90555 


34 




27 


76342 85247 14753 91096 


77387 86842 13158 90546 


33 




28 


76360 85273 14727 91087 


77405 86 868 13132 90537 


32 




29 


76378 85300 14700 91078 


77422 86894 13106 90527 


31 




30 


T.7639S 1.85327 0.14673 T.91069 


7.77439 T.86 921 0.13079 T.90 518 


30 




31 


76413 85354 14646 91060 


77456 86947 13053 90509 


29 




32 


76431 85380 14620 91051 


77 473 86974 13026 90499 


28 




33 


76448 85407 14593 91042 


77490 87000 13000 90490 


27 




34 


76466 85434 14566 91033 


77507 87027 12973 90480 


26 




35 


1.76484 1.85460 0.14540 1.91023 


T.77S24 T.87053 0.12947 1.90471 


25 




36 


76501 85487 14513 91014 


77541 87079 12921 90462 


24 




37 


76519 85514 14486 91005 


77558 87106 12894 90452 


23 




38 


76537 85540 14460 90996 


77 575 87132 12868 90443 


22 




39 


76554 85567 14433 90987 


77592 87158 12842 90434 


21 




40 


T.76 572 1.85 594 0.14 406 1 .90 978 


T.77609 T.8718S 0.12815 T.90 424 


20 




41 


76 590 85 620 14 380 90 969 


77626 87211 12789 90415 


19 




42 


76607 85647 14353 90960 


77643 87238 12762 90405 


18 




43 


76625 85674 14326 90951 


77660 87264 12736 90396 


^l 




44 


76 642 85 700 14 300 90 942 


77677 87290 12710 90386 


16 




45 


T.76 660 T.85727 0.14273 1.90933 


T.77694 T.87317 0.12683 T.90 377 


15 




46 


76677 85754 14246 90924 


77711 87343 12657 90368 


14 




47 


76695 85780 14220 90915 


77728 87369 12631 90358 


13 




48 


76712 85807 14 193 90906 


77744 87396 12604 90349 


12 




49 


76730 85834 14166 90896 


77761 87422 12578 90339 


11 




50 


7.76 747 7.85 860 0.14 140 T.90 887 


T.77778 T.87448 0.12552 T.90 330 


10 




51 


76765 85887- 14 113 90878 


77 795 87475 12525 90320 


9 
8 




52 


76782 85913 14087 90869 


77812 87501 12499 90311 




S3 


76 800 85 940 14 060 90 860 


77829 87527 12473 90301 


7 
6 




54 


76817 85967 14033 90851 


77846 87554 12446 90292 




55 


1.7683s 1.85993 0.14007 1.90842 


T.77862 1.87580 0.12420 1.90282 


5 




56 


76852 86020 13980 90832 


77879 87606 12394 90273 


4 




57 


76870 86046 13954 90823 


77896 87633 12367 90263 


3 




58 


76887 86073 13927 90814 


77913 87659 12341 90254 


2 




^g' 


76904 86100 13900 90805 
T.76 922 1.86 126 0.13874 1.90796 


77930 87685 12 315 90244 


0' 

t 




T.77946 1.87711 0.12289 1.90235 

Incp nn<i loff cot loff tan loff sin 






log 003 log oot log tan log sin 

64° C 


5) 63° "-""g' 


51N, 
3°-66 


etc 

° 



37°-40° 
LOG SIN, etc. 37° 



38° 



/ 


log sin 


log tan 


log oot 


log 003 


log sin 


log tan 


log oot 


log oos 




0' 


1.77 946 


T.87711 0. 


12289 


1.90 235 


T.78 934 


T.89 281 


o.io 719 


T.89 653 


60 


I 


77963 


87738 


12 262 


90225 


78950 


89307 


10693 


89643 


59 


2 


77980 


87764 


12236 


90216 


78967 


89333 


10667 


89633 


58 


3 


77 997 


87 790 


12 210 


90206 


78983 


89359 


10 641 


89624 


57 


4 


78013 


87817 


12 183 


90197 


78999 


89385 


10615 


89614 


S6 


5 


1.78030 


1.87843 0. 


12 157 


1.90 187 


1-79015 


1-89 41 1 


0.10589 


T.89 604 


55 


6 


78047 


87869 


12 131 


90178 


79031 


89437 


10563 


89594 


54 


7 


.78 063 


87895 


12 105 


90168 


79047 


89463 


10537 


89584 


S3 


8 


78080 


87922 


12078 


90159 


79063 


89489 


10511 


89574 


52 


9 


78097 


87948 


12052 


90149 


79079 


89515 


10485 


89564 


SI 


10 


1-78113 


1.87974 0. 


[2026 


T.90 139 


1.79095 


1.89 541 


0.10459 


1.89554 


50 


II 


•78 130 


88000 


t2O0O 


90 130 


79 III 


89567 


10433 


89544 


49 


12 


78147 


88027 


11973 


90 120 


79 128 


89593 


10407 


89534 


48 


13 


78163 


88053 


II 947 


90 III 


79144 


89 619 


10381 


89524 


47 


14 


78180 


88079 


II 921 


90 lOI 


79 160 


89645 


1035s 


89514 


46 


IS 


r.78 197 


T.88 105 0. 


11895 


T.90 091 


T.79 176 


1.89 671 


0.10329 


1.89504 


45 


i6 


78213 


88 131 


II 869 


90082 


79192 


89697 


10303 


89495 


44 


17 


78 230 


88158 


[1 842 


90072 


79208 


89723 


10277 


89485 


43 


18 


78246 


88184 


II 816 


90063 


79224 


89749 


10 251 


8947s 


42 


19 


78263 


88210 


[1 790 


90053 


79240 


89775 


10225 


89465 


41 


20 


T.78 280 


T.88 236 0. 


II 764 


T.90 043 


T.79 256 


1.89 801 


0.10 199 


1-89455 


40 


21 


78296 


88262 


■1738 


90034 


79272 


89827 


10 173 


89445 


39 


22 


78313 


88289 


.1711 


90024 


79288 


89853 


10 147 


8943s 


38 


23 


78329 


88315 


II 685 


90014 


79304 


89879 


10 121 


89425 


37 


24 


78346 


88341 


[1 659 


90005 


79319 


89905 


10095 


89415 


36 


25 


1.78362 


1.88367 0. 


"633 


T.89 995 


1-79 335 


1-89931 


0.10069 


1.89405 


35 


26 


78379 


f!393 


[I 607 


89985 


79351 


89957 


10043 


8939s 


34 


27 


78395 


88420 


[1 580 


89976 


79367 


89983 


10 01 7 


89385 


33 


28 


^^'^"S 


88446 


"554 


89966 


79383 


90009 


09991 


89375 


32 


29 


78428 


88472 


[I 528 


89956 


79 399 


90035 


09965 


89364 


31 


30 


7.78445 


T.88 498 0. 


[I 502 


T.89 947 


1-79 41S 


T.90 061 


0.09 939 


1-89354 


30 


31 


78461 


88524 


[I 476 


89937 


79 431 


90086 


09914 


89344 


29 


32 


78478 


88550 


[I 450 


89927 


79447 


90 H2 


09888 


89334 


28 


33 


78494 


f!577 


[I 423 


89918 


79463 


90138 


09862 


•89 324 


27 


34 


78510 


88603 


■1397 


89908 


79478 


90 164 


09836 


89314 


26 


35 


1.78527 


T.88 629 0. 


«37i 


1.89898 


1.79494 


T.90 190 


0.09 810 


1.89304 


25 


36 


78543 


88655 


II 345 


89888 


79510 


90 216 


09784 


89294 


24 


37 


78560 


8»68i 


I 319 


89879 


79526 


90242 


09758 


89284 


23 


38 


78576 


88707 


1293 


89869 


79542 


90268 


09732 


89274 


22 


39 


78592 


88733 


I 267 


89859 


79558 


90294 


09706 


89264 


21 


40 


1.78609 


1.88759 0. 


I 241 


1.89849 


1-79 573 


T.90 320 


0.09 680 


T.89 254 


20 


41 


78625 


88 786 1 


I 214 


89840 


79589 


90346 


09654 


89244 


19 


42 


78642 


88812 1 


I 188 


89830 


79605 


90371 


09 629 


89233 


18 


43 


78658 


88 838 1 


I 162 


89820 


79621 


90397 


09603 


89223 


17 


44 


78674 


88 864 1 


I 136 


89810 


79636 


90423 


09 577 


89213 


16 


45 


1.78 691 


T.88 890 0. 


I no 


T.89 8oi 


1.79652 


1-90449 


0.09551 


1.89 203 


15 


46 


78707 


88916 1 


1084 


89791 


79668 


90475 


09525 


89193 


14 


47 


78723 


88 942 ] 


1058 


89781 


79684 


90501 


09499 


89183 


13 


48- 


78739 


88 968 1 


I 032 


89771 


79699 


90527 


09473 


89173 


12 


49 


78756 


88 994 J 


I 006 


89761 


79715 


90553 


09447 


89162 


II 


50 


1.78772 


T.89 020 0.1 


0980 


T.89 752 


1-79 731 


1.90578 


0.09 422 


T.89 152 


10 


51 


78788 


89 046 1 


0954 


89742 


79746 


90604 


09396 


89142 


9 


52 


78805 


89073 1 


0927 


89732 


79762 


90630 


09370 


89132 


8 


S3 


78821 


89 099 1 


0901 


89722 


79778 


90656 


09344 


89 122 


7 


54 


78837 


89 125 I 


0875 


89712 


79 793 


90682 


09318 


89 112 


6 


55 


1.78853- 


'.89151 0.1 


0849 


1.89 702 


1.79809 


T.90 708 


0.09 292 


T.89 101 


5 


56 


78869 


89 "77 I 


0823 


89693 


79 825 


90734 


09 266 


89091 


4 


57 


78886 


89 203 1 


0797 


89683 


79840 


90759 


09241 


89081 


3 


58 


78902 


89 229 I 


0771 


89673 


79856 


90785 


09215 


89071 


2 


59 


78918 


89 255 I 


°745 


89663 


79872 


90 81 1 


09189 


89060 


1 


60' 


1-78934 " 


". 89 281 0.1 


0719 


1.89653 


1.79887 


1.90837 


0.09 163 


1.89050 


0' 




log 00s 


log cot ] 


og tan 


log sin 


log 003 


log oot 


log tan 


log sin 


t 



LOG SIN, etc. fiO° 

49°-62° 



(66) 



6r 



^^__ 


39° 


4U LOG SIN. etc 


" fir 


log sin log tan log cot log 00s 


log sin log tan log oot log 003 






0' 

I 

2 


1.79887 1-90837 0.09163 1.89050 
79903 90863 09137 89040 
79918 90889 09 I II 89030 


1.80807 1.92381 0.07619 7.88425 
80822 92407 07593 88415 
80837 92433 07567 88404 
80852 92458 07542 88394 

_ 80 867 92484 07516 88383 


60' 

It 

11 
55 
54 
53 
52 
51 




3 


79 934 90914 09086 89020 




4 


79950 90940 09060 89009 




S 


1.79965 1.90966 0.09034 1.88999 


1.80882 1.92 5 10 0.07490 1.88372 
80897 92535 07465 88362 
80912 92561 07439 88351 
80927 92587 07413 88340 




6 


79981 90992 09008 88989 




7 


79996 91 018 08982 88978 




8 


80012 91043 08957 88968 




9 


80027 91069 08931 88958 


80942 92612 07388 88330 




10 


T.80043 T.9109S 0.08905 T.88948 


7.80957 7.92638 0.07362 7.88319 


50 




II 


80058 91 121 08879 88937 


80972 92663 07337 88308 


49 




12 


80074 91 147 08853 88927 


80987 92689 07311 88298 


48 




13 


80089 91 172 08828 88917 


81002 92715 07285 88287 


47 




14 


80105 91 198 08802 88906 


81 017 92740 07260 88276 


46 




ij 


T.So 120 r.91 224 0.08776 T.88896 


1. 8 1 032 1.92766 0.07234 7.88266 


45 




i6 


80136 91250 08750 88 886 


81047 92792 07208 88255 


44 




17 


80 151 91 276 08 724 88 87s 


81 061 92817 07183 88244 


43 




18 


80 166 91 301 08 699 88 865 


81076 92843 07157 88234 


42 




«9 


80182 91327 08673 8885s 


81 091 92 868 07 132 88 223 


41 




20 


T.80197 T.91353 0.08647 T.88844 


7.81 106 7.92 894 0.07 106 7.88 212 


40 




21 


80 213 91 379 08 621 88 834 


81 121 92920 07080 88201 


39 




22 


80228 91404 08596 88824 


81 136 92945 07055 88 191 


38 




23 


80244 91430 08570 88813 


81 151 92971 07029 88180 


37 




24 


80259 91456 08544 88803 


81 166 92996 07004 88169- 


36 




25 


7.80274 T.91482 0.08518 1.88793 


1.81 180 7.93022 0.06978 7.88158 


35 




26 


80290 91507- 08493 88782 


81 195 93 048 06 952 88 148 


34 




''^ 


80305 91533 08467 88772 


81 210 93073 06927 88137 


33 




28 


80320 91559 08441 88761 


81225 93099 06901 88126 


32 




29 


80336 91585 08415 88751 


81240 93124 06876 88 115 


31 




30 


1.80351 1.91610 0.08390 1.88 741 


7.81254 7.93150 0.06850 7.88105 


30 




31 


80366 91636 08364 88730 


81269 93175 06825 88094 


29 


> 


32 


80 382 91 662 08 338 88 720 


81 284 93 201 06 799 88 083 


28 




33 


80 397 91 688 08 312 88 709 


81299 93227 06773 88072 


27 




34 


80 412 91 713 08 287 88 699 


81314 93252 06748 88061 


26 




35 


T.80428 T.91 739 0.08261 T.88 688, 


1. 8 1 328 1.93278 0.06722 7.88051 


25 




36 


80443 91765 08235 88678 


81343 93303 06697 88040 


24 




27 


80458 91791 08209 88 668 


81358 93329 06671 88029 


23 




38 


80473 91 816 08184 88657 


81372 93 354 06646 88018 


22 




39 


80489 91842 08158 88647 


81387 93380 06620 88007 


21 




40 


T.80504 T.91868 0.08132 T.88636 


7.81402 7.93406 0.06594 7.87996 


20 




4J 


80519 91893 08107 88626 


81417 93431 06569 87985 


19 




42 


80534 91919 08081 88615 


81 431 93 457 06543 87975 


18 




43 


80 550 91 945 08 055 88 605 


81446 931^82 06518 87964 


17 




44 


80565 91 971 08029 88594 


81 461 93508 06492 87953 


16 




4| 


1.80580 1. 91 996 0.08004 T.88584 


1.81475 1-93 533 0.06467 7.87942 


15 




46 


80595 92022 07978 88573 


81490 93 559 06441 87931 


14 




'^l 


80610 92048 07952 88563 


81505 93584 06416 87920 


13 




48 


80625 92073 07927 88552 


81 519 93610 06390 87909 


12 




49 


80641 92099 07901 88542 


81 534 93 636 06 364 87 898 . 


II 




50 


T.80656 T.92125 0.07875 7.88531 


7.81 549 7.93 661 0.06 339 7.87 887 


10 




SI 


80671 92150 07850 88521 


81563 93687 06313 87877 


9 




52 


80686 92176 07824 88510 


81578 93712 06288 87866 


8 




53 


80701 92202 07798 88499 


81 592 93 738 06 262 87 855 


7 




54 


80716 92227 07773 88489 


81 607 93 763 06 237 87 844 


6 




5| 


1.80 731 1.92253 0.07747 7.88478 


1.81622 1.93789 0.06211 1.87833 


5 




56 


80746 92279 07721 88468 


81 636 93 814 06 186 87 822 


4 




57 


.80762 92304 07696 88457 


81651 93840 06160 8y8ii 


3 




58 


80777 92330 07670 88447 


81665 93865 06135 87800 


2 




59 


80 792 92 356 07 644 88 436 


81 680 93 891 06 109 87 789 


I 




60' 


1.80807 1.92 381 0.07619 1.88425 


1.81694 1.93 916 0.06084 1-87778 


0' 






log cos log cot log tan log sin 


log 00s log oot log tan log sin 


r 




- - ■ 


ao° ^^ 


7) 49° LOG 1 


N, e 
°-62° 


tc. 



41°-44° 
LOG SIN, etc. 4r 



42° 



/ 


log sin 


log tan 


log cot 


log 00s 


log sin 


log tan 


log cot 


log cos 




0' 


1. 81 694 


r.93916 


0.06 084 


T.87 778 


T.82 551 


1-95 444 


0.04 556 


T.87 107 


60' 


I 


81709 


93942 


06058 


87767 


82565 


95469 


04531 


87096 


59 


2 


81 723 


93967 


06033 


87756 


82579 


95 495 


04505 


87085 


58 


3 


81738 


93 993 


06007 


8774s 


82593 


95520 


04480 


87073 


57 


4 


81752 


94018 


05 982 


87734 


82607 


95 545 


04455 


87062 


56 


S 


1. 81 767 


T.94 044 


0.05 956 


1.87 723 


T.82 621 


1-95571 


0.04 429 


T.87 050 


55 


6 


81 781 


94069 


05931 


87712 


82635 


95596 


04404 


87039 


54 


7 


81 796 


94095 


05905 


87701 


82649 


95622 


04378 


87028 


53 


S 


81 810 


94 120 


05880 


87 690 


82663 


95647 


04353 


87016 


52 


9 


81825 


94146 


05854 


87679 . 


82677 


95672 


04328 


87005 


51 


10 


T.81 839 


T.94 171 


0.05 829 


T.87 668 


1.82 691 


T.95 698 


0.04 302 


1.86993 


50 


II 


81854 


■94197 


05803 


87657 


82705 


95723 


04277 


86982 


49 


12 


81868 


94222 


05778 


87646 


82719 


95748 


04252 


86970 


48 


13 


81882 


94248 


05752 


87635 


82733 


95 774 


04226 


86959 


47 


14 


81897 


94273 


05727 


87624 


82747 


95 799 


04201 


86947 


46 


'S 


1. 81 911 


1.94299 


0.05 701 


1.87 613 


1.82 761 


1-95 825 


0.04175 


1.86936 


45 


i6 


81926 


94324 


05 676 


87601 


82775 


95850 


04 150 


86924 


44 


17 


81 940 


94350 


05 650 


87590 


82788 


95875 


04125 


86913 


43 


18 


81955 


94 375 


05625 


87579 


82802 


95901 


04099 


86902 


42 


19 


81969 


94401 


05599 


87568 


82816 


95926 


04074 


86890 


41 


20 


T.81 983 


T.94 426 


0.05 574 


1-87 557 


T.82 830 


1-95952 


0.04 048 


T.86 879 


40 


21 


81998 


94452 


05548 


87546 


82844 


95 977 


04023 


86867 


39 


22 


82012 


94 477 


05523 


87535 


82858 


96002 


03998 


86855 


38 


23 


82026 


94503 


05497 


87524 


l^lV 


96028 


03972 


86844 


37 


24 


82041 


94528 


05472 


87513 


82885 


96053 


03947 


86832 


36 


25 


T.82 055 


1.94 554 


0.05 446 


1.87 501 


1.82899 


1.96078 


0.03 922 


1.86821 


35 


26 


82069 


94 579 


05421 


87490 


82913 


96 104 


03896 


86809 


34 


27 


82084 


94604 


05396 


87479 


82927 


96 129 


03871 


86798 


33 


28 


82098 


94630 


05370 


87468 


82941 


96155 


03845 


86786 


32 


29 


82 112 


9465s 


05345 


87457 


82955 


96180 


03820 


86775 


31 


30 


T.82 126 


T.94 681 


0.05 319 


1.87446 


1.82968 


T.96 205 


0.03 795 


1.86763 


30 


31 


82 141 


94706 


05294 


87434 


82982 


96231 


03769 


86752 


29 


32 


8215s 


94732 


05268 


87423 


82996 


96256 


03744 


86740 


28 


33 


82169 


94 757 


05243 


87412 


83010 


96 281 


03719 


86728 


27 


34 


82184 


94783 


05217 


87401 


83023 


96307 


03693 


86717 


26 


35 


1.82 198 


T.94 808 


0.05 192 


T.87 390 


.1.83037 


1.96332 


0.03 668 


T.86 705 


25 


36 


82212 


94834 


05 1 66 


87378 


83051 


96357 


03643 


86694 


24 


37 


82226 


94859 


05141 


87367 


83065 


96383 


03617 


86682 


23 


38 


82240 


94884 


05 116 


87356 


83078 


96408 


03592 


86670 


22 


39 


8225s 


94910 


05090 


87345 


83092 


96433 


03567 


86659 


21 


40 


1.82 269 


T-94 935 


0.05 065 


1.87334 


T.83 106 


1-96459 


0.03 541 


T.86 647 


20 


41 


82283 


94961 


05039 


87322 


83 120 


96484 


03516 


86635 


19 


42 


82297 


94986 


05014 


87311 


83133 


96510 


03490 


86624 


18 


43 


82 311 


95 012 


04988 


87300 


83J47 


96535 


03465 


86612 


'? 


44 


82326 


95037 


04963 


87288 


83161 


96560 


03440 


86600 


l6 


45 


1.82340 


1.9s 062 


0.04 938 


1.87277 


T.83 174 


T.96 586 


0.03 414 


1.86589 


15 


46 


82354 


95088 


04912 


87266 


83188 


96 611 


03389 


86577 


14 


47 


82368 


95 "3 


04887 


87255 


83 202 


96636 


03364 


86565 


13 


48 


82382 


95139 


04861 


87243 


83 215 


96662 


03338 


86554 


12 


49 


82396 


95164 


04836 


87232 


83229 


96687 


03313 


86542 


II 


50 


1.82 410 


T.95 190 


0.04 810 


T.87 221 


T.83 242 


T.96 712 


0.03 288 


T.86 530 


10 


51 


82424 


95 215 


04785 


87 209 


83256 


96738 


03 262 


86518 


9 


52 


82439 


95240 


04 760 


87198 


83270 


96763 


03 237 


86507 


8 


53 


82453 


95 266 


04734 


87187 


83283 


96788 


03 212 


86495 


7 


54 


82467 


95291 


04709 


87>7S 


83297 


96 814 


03186 


86483 


6 


55 


1.82 481 


1-95317 


0.04 683 


1.87 164 


1.83 310 


T.96 839 


0.03 161 


1.86472 


5 


56 


82495 


95342 


04658 


87153 


83324 


96864 


03136 


86460 


4 


57 


82509 


95368 


04632 


87 141 


83338 


96890 


03 no 


86448 


•3 


58 


82523 


95 393 


04607 


87130 


83351 


96915 


03085 


86436 


2 


59 


82537 


95418 


04582 


87119 


83365 


96940 


03060 


86425 


1 


60' 


1.82 551 


1.95444 


0.04 556 


1.87 107 


1-83378 


1.96966 


0.03 034 


1.86413 


0' 




log cos 


log cot 


log tan 


logain 


log cos 


log cot 


log tan 


log sin 


t 



LOG SIN, etc. ILK" 

45°-48° 



(68) 



47° 



43^^ 



41°-44° 
44° LOG SIN, etc. 



9 


log sin 


log tan 


log cot 


log cos 


log sin 


log tan log cot log cos 




0' 


7.83 378 


T.96 966 


0.03 034 


T.86413 


T.84 177 


1.98484 0.01 516 T.85 693 


60' 


I 


83392 


96991 


03009 


86401 


84190 


98 509 01 491 85 681 


59 


2 


83405 


97016 


02984 


86389 


84203 


98 534 01 466 85 669 


58 


3 


83419 


97042 


02958 


86377 


84216 


98 560 01 440 85 657 


57 


4 


83432 


97067 


02933 


86366 


84229 


98585 01415 85645 


56 


5 


1.83446 


1.97092 


0.02 908 


1.86354 


T.84 242 


1.98 610 O.OI 390 1.85632 


55 


6 


83459 


97118 


02882 


86342 


84255 


98 635 oi 365 85 620 


54 


7 


83473 


97143 


02857 


86330 


84269 


98 66i 01 339 85 608 


53 


8 


83486 


97168 


02832 


86318 


84282 


98686 01 314 85596 


52 


9 


83500 


97193 


02807 


86306 


84295 


98711 01289 85583 


51 


10 


1-83513 


^.97 219 


0.02 781 


T.86 295 


T.84 308 


T.98 737 0.01 263 T.85 571 


50 


II 


83527 


97244 


- 02756 


86283 


84321 


98 762 01 238 85 559 


49 


12 


83540 


97269 


02731 


86271 


84334 


98 787 01 213 85 547 


48 


13 


83554 


97295 


02 705 


86259 


84347 


98812 01 i88 85 534 


47 


14 


83567 


97320 


02 680 ■ 


86247 


84 360 


98838 01162 85522 


46 


15 


1.83581 


1-97 345 


0.02 655 


T.86 235 


1-84 373 


1.98863 0.01 137 T.85 510 


45 


16 


f3 594 


97371 


02629 


86223 


84385 


98888 01112 85497 


44 


17 


83608 


97396 


02604 


86211 


84398 


98 913 01 087 85 485 


43 


18 


83621 


97421 


02579 


86 200 


84411 


98 939 01 061 85 473 


42 


19 


83634 


97 447 


02553 


86188 


84424 


98 964 01 036 85 460 


41 


20 


1.83648 


T.97 472 


0.02 528 


T.86 176 


T.84 437 


T.98 989 o.oioii T.85 448 


40 


21 


83661 


97 497 


02503 


86164 


84450 


99015 00985 85436 


39 


22 


83674 


97523 


02477 


86152 


84463 


99 040 00 960 85 423 


38 


23 


83688 


97548 


02452 


86 140 


84476 


99065 00935 85411 


37 


24 


83701 


97 573 


02427 


86128 


84489 


99090 00910 85399 


36 


^1 


'o^ 7'S 


1.97598 


0.02 402 


1.86116 


1 .84 502 


1.99116 0.00884 T.85 386 


35 


26 


83728 


97624 


02376 


86 104 


84515 


99 141 00 859 85 374 


34 


27 


83741 


97649 


02351 


86092 


84528 


99166 00834 85361 


a 


28 


i3 7SS 


97674 


02326 


86080 


84540 


99 191 00809 85349 


32 


29 


8-3768 


97700 


02 300 


86068 


84553 


99217 00783 85337 


31 


30 


1.83 781 


T.97 725 


0.02 275 


T.86 056 


1.84566 


1.99242 0.00758 T.85 324 


30 


3" 


83795 


97750 


02250 


86044 


84579 


99267 00733 85312 


29 


32 


83808 


97776 


02 224 


86032 


84592 


99 293 00 707 85 299 


28 


33 


83821 


97 801 


02 199 


86020 


84605 


99 318 00 682 85 287 


27 


34 


83834 


97826 


02 174 


86008 


84618 


99 343 00 657 85 274 


26 


35 


T.83 848 


7.97851 


0.02 149 


T.85 996 


T.84 630 


1.99368 0.00632 1.85262 


25 


36 


83861 


97877 


02 123 


85984 


84643 


99 394 00 606 85 250 


24 


37 


83874 


97902 


02098 


85972 


84656 


99419 00581 85237 


23 


38 


83887 


97927 


02073 


85960 


84669 


99 444 00 556 85 225 


22 


39 


83901 


97 953 


02047 


85948 


84682 


99469 00531 85212 


21 


40 


1.83914 


T.97 978 


0.02 022 


1-85936 


T.84 694 


1^-99 495 0.00 505 T.85 200 


20 


41 


83927 


98003 


01997 


85924 


84707 


99520 00480 85187 


19 " 


42 


83940 


98029 


01971 


85912 


84 720 


99 545 00455 85175 


18 


43 


83954 


98054 


01 946 


85 900 


84733 


99 570 00430 85 162 


«7 


44 


83967 


98079 


01 921 


85888 


84745 


99 596 00404 85 150 


16 


45 


1.83980 


1.98 104 


o.oi 896 


T.85 876 


T.84 758 


T.99 621 0.00 379 T.85 137 


15 


46 
47 


83993 


98130 


01 870 


85864 


84771 


99 646 00 354 85 125 


14 


. 84006 


98155 


01 845 


85851 


84784 


99 672 00328 85 112 


13 


48 


84020 


98180 


01 820 


85839 


84796 


99 697 00 303 85 100 


12 


49 


84033 


98206 


01794 


85827 


84809 


99 722 00 278 • 85 087 


11 


50 


1.84046 


T.98 231 


0.01 769 


1.85815 . 


T.84 822 


T.99 747 0.00253 T.85 074 


10 


51 


84059 


98256 


01 744 


85 803 


84835 


99 773 00 227 85 062 


9 


52 


84072 


98281 


oi 719 


85791 


84847 


99 798 00 202 85 049 


8 


53 


84085 


98307 


01693 


85779 


84860 


99823 00177 85037 


7 


54 


84098 


98332 


01668 


85766 


84873 


99848 00152 85024 


6 


55 


T.84 112 


1-98357 


0.01 643 


1.85754 


1.84885 


T.99 874 0.00126 T.85 012 


5 


56 


84125 


98383 


01 617 


85742 


84898 


99 899 00 101 84 999 


4 


57 


84138 


98408 


01 592 


85730 


84911 


99 924 00 076 84 986 


3 


58 


84 151 


98433 


01567 


85718 


84923 


99949 00051 84974 


2 


59 


84 164 


98458 


01542 


85706 


84936 


1-99 975 00025 84961 


I 


60' 


1.84177 


1 .98 484 


0.01 516 


1.85693 


1.84949 


0.00000 0.00000 1.84949 


0' 




log cos 


log cot 


log tan 


log sin 


log cos 


log cot log tan log sin 


t 



w 



(69) 



450 LOG SIN, etc. 
45°-48° 



CONSTANTS. 



CONSTANTS. 



MATHEMATICAL CONSTANTS. 


Quantity. 


Numerical 




Common 




Talue. 




LOGABITHM. 




2.71 828 18 


Base of Napierian, natural, or hyperbolic 
logarithms. 


0.43 429 45 


l/lOgl0£ 


2.30 258 5 


Factor to multiply into comTnon logs to 
convert into Napierian logs. 


0.36 221 57 


lOgioe 


0434294s 


Factor to multiply into Napierian logs to 
convert into common logs. 


7.6377843 


ic 


3.14 159 26s 


Ratio of circumference to diameter. 


0.4971499 


ir! 


9.86 960 44 


Square of ir. 


0.99 429 97 


I radian 


57° 17' 45" 


57.° 29 58 = 206265." = arc equal to 
radius. 






UNITED STA1 


•ES, BRITISH, AND METRIC UNITS. 




Note. — Th 


e foUowingr ratios are 


given on the authority of the U. S. Coast and Geodetic Surrey, | 


"Tables of W 


sights and Measures. 


Washington, D. C, 1890." 




I metre 


39.37 inches. 


This is the legalized ratio for the U. S. 
The U. S. and the British inch are equal. 
By comparisons to date (July, 1895), it 
appears probable that this value is smal- 
ler than the real ratio of the "Metre des 


1-5951654 






Archives ' ' to the thirty-sixth part of the 








"Imperial Standard Yard" by one or 








two parts in one million. 




I metre 


1.09 361 I yard. 


The U. S. and the British yard are 


0.03 886 29 






equal. 


• 


I metre 


3.28 08 33 feet. 




0.5159842 


I kilometer 


0.62 13 70 mile 


of 5280 feet. 


1-7933503 


I mile 


1.60 934 7 kilom. 




0.20 664 97 


I yard 


0.91 440 2 metre. 




T.9611371 


I foot 


0.30 480 I metre. 




7.48 401 58 


I inch 


25.40 005 mm. 


Deduced from above legalized ratio of 
yard and metre in TJ. S. 


1.4048346 


zinch 


25.40 000 mm. 


is more convenient besides being proba- 
bly more exact. It is probably about 
one part in one million too small, as 


1.4048337 






the reciprocal of 0.0254 is 39.37 008. 





CONSTANTS. 



(70) 



CONSTANTS. 



QUASTITT. 



I pound Av. 



I pound Av. 
1 ounce Av. 
I ounce Troy 
I grain 

I kilogramme 
I gramme 
I litre 



I litre 
I litre 

I quart, XJ. S. 
I gallon, U. S. 
I fluid ounce 
I bushel, U. S. 
I British gallon 
I British bushel 



numekioal 
Talhe. 



7000 grains. 



453-59 242 77 grammes. 
28.34 953 grammes. 
31.10348 grammes. 
0.06 479 892 gramme. 

2.20 462 2 pounds Av. 
1543 235 639 grains. 
1.05 668 U. S. quarts. 



0.26417 U. S. gallon. 
33.814 U. S. fluid oz. 
0.94 636 litre. 
3.78 544 litres. 
0.02 957 3 litre. 
231 cu. inches. 
4.54 346 litres. 
36.34 77 litres. 



CONSTANTS. 



The pound avoirdupois is the 
same in Great Britain and the 

U.S. 



Avoirdupois and Troy grains are 
the same. 



By original definition one litre 
was the volume of one cubic 
decimetre, but at present the 
accepted definition is that pro- 
visionally adopted by the Inter- 
national Bureau of Weights and 
Measures in 1880, viz. ; the 
volume of one kilogramme of 
water at its maximum density. 
The experimental determina- 
tion with high accuracy of the 
relation between this volume 
and the cubic decimetre is still 
unfinished. The following val- 
ues assume this ratio to be 
unity. 

ZZ 

7 



Common 

LOGAllITHM. 



3.84 509 80 



2.65 666 58 
1.45 254 59 
1.49 280 91 
2.81 156 78 

0-34 333 42 
1. 18 843 22 
0.023944 



1.42 1884 
1.52 910 
T.97 605 6 
0.57 811 6 
2.47 090 
2.36 361 20 
0-65 738 67 
1.5604769 



MECHANICAL OR DYNAMICAL EQUIVALENT OF HEAT. 

The best values of this quantity (usually denoted by J') at present attainable (Novem- 
ber, 1895) are the following. The values are uncertain by only about ± one twentieth 
of one per cent. 
427.3 kilogrammetres of work or energy are required at latitude 45°, sea-level 

{g = 980.6 c.g.s.), to raise i kilogramme of water through 1° C. at 15° C. 
778.8 ft. lbs. of work or energy are required at latitude 45"-", sea-level {g = 980.6 c.g.s.), 

to raise i lb. of water through i" Tahr. at 59° Fahr. (= 15° C). For most 

engineering purposes 779 ft. lbs. would be near enough. 



(70 



CONSTANTS. 



CONSTANTS. 



CONSTANTS. 



140a ft. lbs. of work or energy are required at latitude 45°, sea-level (g = 980.6), to 
raise i lb. of water througb 1° Cent, at 15° C. For most engineering purposes 
1400 ft. lbs. would be near enough. 
4.i90"io' ergs of work or energy are required to raise i gramme of water through 1° C. 
at 15° C. 
To reduce these values to any given locality, multiply by the ratio gu '■ Ci where gis is 
the value (980.6) of the acceleration of gravity at latitude 45°, sea-level, and g is the 
value at the given place. The latter may be obtained from the latitude and altitude of 
the place by the formula given upon the next page, unless otherwise better known. The 
altitude correction is but six one-thousandths of one per cent (0.00 006) for each 1000 ft. 
of elevation, and therefore quite negligible. Within the limits of uncertainty of the 
quantities involved the latitude correction for places between 30° and 60° may be 
applied thus : — 

778.8 



For each degree of latitude north of 45° subtract 
For each degree of latitude south of 45" add 



427-3 

0.04 kgm. 

0.04 kgm. 



0.07 ft. lbs. 
0.07 ft. lbs. 



1402 

0.13 ft. lbs. 

0.13 ft. lbs. 



Note. — The persistence with ■which the time-honored values, 772 ft. lbs. and 424 fegm., of this most impor- 
tant constant are adhered to in practice, although known to be nearly one per cent too small, is due largely to 
the flagrant negligence of the authors of text-books of both physics and engineering. No attention is paid to 
the fact that Joule's original data have been amended acceptaMy to Mm, and that his work has been supple- 
mented by the elaborate researches of at least three other independent observers with radically diverse 
methods. How remarkably these new results check each other and confirm Joule's amended results may be 
seen from the following table, which is given to indicate the source of the foregoing values. 





OEiapjAL Data. 


Reduced 
TO Lat. 
45° Sea- 
Level. 


D1FF8. 

FROM 

Mean. 


Bate. 




ATTTHOErrv. 


J. 
kgm. 


ff- 


t°.C. 


Refeeehoe. 


JoTJLE (as correclied and 














See quotations in the 


reduced to Baltimore by 














Rowland and GrifSths 


Rowland.) [Assigning eq. 














referenceSo 


wts. to all methods.] 


427.99 


980.05 


iS-° 










[Assigning Rowland's arbi- 
















trary wts.] 


426.66 


980.05 


iS-° 










Mean of both. 


427-33 


980.05 


i5-° 


427.08 


-.16 


1847-7B 




Rowland [at Baltimore]. 


427.4 


980.05 


15-° 


427.16 


-.08 


1879 


Proc. Am. Acad. A. 
and S. XV. 75 (1880). 


Gbutiths [at Greenwichl. 


4.194.10' 


981.17 


l5-° 


427.70 


-I-.46 


1892 


Phil. Transac. cbcxxiv. 


(In ergs.) 














496 (1893). 


MiOFLESOlJ [at Paris]. 


426.84 


981.00 


15-° 


427.01 


-23 


1892 


Ann. de Ch. et de Ph. 


- 














xxvii. 237 (1892). 


Mean of all. 








427.24 


±.23 






Mean omitting Joule. 


427.29 













The specific heat of water, and therefore the value of J, diminishes slightly with rise 
of temperature. The rate of this diminution is not yet satisfactorily determined, but 
about as nearly as it is now known the true specific heat St at any temperature t° not far 
from 15° C, may be expressed in terms of true specific heat su at 15° C. by 

St = S16 [i. - 0.00 030 (S - 15)]. 
Hence Jt, the number of kgm. or ergs necessary to raise i kgr. of water from t° to 
t° + 1°, will be 

J, = Ju [I. - 0.00 030 (t - 15)] [Range 13° - 20°], 



CONSTANTS. 



(72) 



CONSTANTS. 

CONSTANTS. 



or for I lb. of water i° Fahr. 

Jt=Jaali- o.ooo3o(t - 59)] [Range 56° - 68°]. 

_ The values of Ji^ and J^a are given on pages 71 and 72. For further discussion of this 
subject consult Griffiths, Phil. Mag. xi. 431 (1895). 

The scale of temperature in which these results are expressed is the hydrogen scale 
of the International Bureau of Weights and Measures, which represents, as nearly as it is 
known, the Thomson absolute scale. 



VALUE OF g AT DIFFERENT LATITUDES AND ELEVATIONS. 

9 = 9iS 0(1 — 0.00 259 cos 2 X — 0.00 000 020 S). 

gufi =■ 980.6 — - approx. This is the approximate average value of g at latitude 45°, 
sec^ 

sea-level. The experimental values vary widely in the next place of figures. 

X = latitude of place. 

H = altitude of place above sear level in metres. 

Note. — Very recent observations render it probable that near the earth's surface the coefficient of R is 
more nearly 0.00 000 030. 



BROWN AND SHARPE WIRE GAUGE. 

The diameter corresponding to any gauge number above zero (i.e. of any size less 
than that of a No. o wire) may be found to within one hundred-thousandth of an inch 
(five decimal places) by the expression 

Diam. in inches of i „. „ 

} = 0.32 486 X 0.89 052 5", 
gauge number m J 

or, Log of diam. in inches =7.51 170 -|- 1.94 964 5 w, 

or, " " " " " — 9.51 170 — 10. -1- (9.94 9645 — io.)n. 

The diameter corresponding to any gauge numbers o, 00, 000, and so on, may simi- 
larly be computed by the following expressions in which N is the number of zeros. 

Diam. in inches = 0.28 930 x 1.12 293^, 
or, Log of diam. in inches = T.46 1348 -|- 0.05 035 3 N. 

The two primary sizes on which the gauge is based are No. 0000, diameter 0.46 inch 
exactly, and No. 36, diameter 0.005 ^^'^^ exactly. 



PHYSICAL AND CHEMICAL CONSTANTS. 

For very reliable and extended tables consult Landolt und Bornstein, Physikalisch — 
Chemische Tabellen. 



(73) CONSTANTS. 



A HISTORY OF 

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